11institutetext: Department of Biomedical Engineering, King’s College London
22institutetext: Centre for the Developing Brain, King’s College London
33institutetext: Lab for AI in Medicine, Technical University of Munich,
Germany

# Surface Masked AutoEncoder: Self-Supervision for Cortical Imaging Data

Simon Dahan 11 Mariana Da Silva 11 Daniel Rueckert 33 Emma C. Robinson 1122

###### Abstract

Self-supervision has been widely explored as a means of addressing the lack of
inductive biases in vision transformer architectures, which limits
generalisation when networks are trained on small datasets. This is crucial in
the context of cortical imaging, where phenotypes are complex and
heterogeneous, but the available datasets are limited in size. This paper
builds upon recent advancements in translating vision transformers to surface
meshes and investigates the potential of Masked AutoEncoder (MAE) self-
supervision for cortical surface learning. By reconstructing surface data from
a masked version of the input, the proposed method effectively models cortical
structure to learn strong representations that translate to improved
performance in downstream tasks. We evaluate our approach on cortical
phenotype regression using the developing Human Connectome Project (dHCP) and
demonstrate that pre-training leads to a 26% improvement in performance, with
an 80% faster convergence, compared to models trained from scratch.
Furthermore, we establish that pre-training vision transformer models on large
datasets, such as the UK Biobank (UKB), enables the acquisition of robust
representations for finetuning in low-data scenarios. Our code and pre-trained
models are publicly available at https://github.com/metrics-lab/surface-
vision-transformers.

###### Keywords:

Transformers Self-supervision Neuroimaging

##  1 Introduction

Vision transformer (ViT) architectures [9] have shown promising results on
non-Euclidean manifolds [7, 17, 24, 15], which pose challenges for Convolution
Neural Networks (CNNs) due to the non-trivial translation of convolution
operations to domains without a global coordinate system [10, 4]. By re-
framing image modelling as a sequence-to-sequence learning problem, Surface
Vision Transformers (SiTs) [7] provide an effective solution for surface-based
domains, increasingly utilised by neuroimaging and cardiac applications [3,
18, 22, 23].

Despite achieving remarkable performances, ViTs mostly rely on using large
training datasets, to compensate for the lack of transformation equivariance
and weight sharing of the model architecture. Therefore, self-supervised
learning (SSL) has been widely explored as a means of addressing this issue.
In the context of ViTs, SSL tasks have drawn inspiration from NLP
counterparts, such as autoregressive models like GPT [21] or Masked Language
Models (MLM) with denoising autoencoders [8], or implementing contrastive
learning strategies that heavily rely on image data augmentation [5]. For
instance, the original ViT [9] adapted the Masked Patch Pretraining (MPP)
method from BERT [8], which involves first corrupting (masking or swapping)
image patches, then training an encoder-decoder architecture to reconstruct
these patches. More recently, [13] demonstrated improved performance with a
Masked AutoEncoder (MAE) strategy that reconstructs sets of masked-out image
patches using only the remaining _unmasked_ patches. The MAE approach exhibits
better image reconstruction and improved performance on downstream tasks
relative to the MPP. Such SSL techniques have also been applied to non-
Euclidean domains. For instance, in the context of irregular 3D meshes, an
adapted MAE has shown promising results in mesh classification and
segmentation [17]. Additionally, MAE pretraining has been successfully
utilised in graph classification tasks, achieving competitive performance in
molecular property predictions [14].

Pre-training with SSL offers particular potential for biomedical applications,
where datasets can be small, and labels are not readily available as they
require manual annotation by medical experts. Indeed, SSL has been broadly
applied for pretraining medical CNNs, in order to take advantage of large-
scale unannotated data [6, 2, 16, 11]. In this work, we therefore propose
Surface Masked AutoEncoder (sMAE) pre-training of Surface Vision Transformers
(SiTs) [7], extending the Masked AutoEncoder (MAE) methodology to the study of
cortical features on regular surface meshes. Our results show that: 1) sMAE
pre-training improves the performance of SiTs on cortical phenotype
regression; 2) Remarkable performance levels are achieved on downstream tasks
even in low-data regimes; and 3) these benefits are comparable whether the
model is pre-trained on the same data used for prediction, or on an
independently collected large dataset (UK Biobank). This suggests the
potential of using sMAE to transfer the benefits of deep learning to much
smaller datasets.

##  2 Methods

###  2.1 Transformer Architecture

The Vision Transformer (ViT) reformulates the image feature learning problem
to a sequence-to-sequence learning paradigm, by regularly patching images into
non-overlapping blocks. Typically, ViTs process images as sequences of
($16\times 16\times 3$ pixels) square image patches and pass them through a
stack of encoder layers, applying self-attention to capture global
dependencies among patches.

Figure 1: sMAE learning pipeline. The sMAE regime involves masking part of the
input token sequence (1.b), before passing the unmasked tokens to the sMAE
encoder (SiT-tiny) (1.c). Unmasked token embeddings and a set of mask tokens
are then concatenated (1.d) and passed to the decoder to reconstruct the
masked patches (1.e). End-to-end finetuning (2) involves extracting the sMAE
encoder and performing supervised training for downstream tasks, by finetuning
or freezing the entire encoder, alongside the MLP head.

Here, we adopt the SiT architecture [7], which translated the concept of ViTs
to sphericalised cortical meshes by representing the cortical surface as a
mesh structure on a sphere. Input data $X\in\mathbb{R}^{|V_{6}|\times C}$ ($C$
channels) is represented on a 6th-order icospheric (ico6) tessellation:
$I_{6}=(V_{6},F_{6})$, with $|V_{6}|=40962$ vertices and $|F_{6}|=81920$
faces. Surface patching is achieved by tessellating ico6 surfaces according to
the regular subdivisions of a low-resolution icosphere, typically ico3
$I_{3}=(V_{3},F_{3})$ ($|F_{3}|=1280$ and $|V_{3}|=642$), see Figure 1.a. This
partition leads to a sequence of $|F_{3}|=1280$ non-overlapping triangular
patches: $T_{3}=\\{t^{1}_{3},t^{2}_{3},..t_{3}^{|F_{3}|}\\}$ (with
$t^{i}_{3}\subset V_{6};|t^{i}_{3}|=|t_{3}|=45$). Imaging features for each
patch are then concatenated across channels, and flattened to produce an
initial sequence of tokens:
$\hat{\mathcal{X}}=\left[X^{0}_{1},...,X^{0}_{N}\right]\in\mathbb{R}^{N\times(|t_{3}|C)}$,
where $N=|F_{3}|$. These are first projected onto a $D-$dimensional sequence
using a trainable linear layer. Then, an extra $D$-dimensional token for
regression or classification is concatenated ($X^{0}_{0}$) and positional
embeddings ($E_{pos\\_enc}\in\mathbb{R}^{(N+1)\times D}$) are added to encode
patch location into a sequence of tokens such that
$\mathcal{X}=\left[X^{0}_{0},...,X^{0}_{N}\right]+E_{pos\\_enc}\in\mathbb{R}^{(N+1)\times
D}$. In contrast to [7], we adopt positional embeddings in the form of 1D
sine-cosine embeddings. Then, this sequence of tokens is processed by the SiT
network which consists of $L$ consecutive transformer encoder blocks of Multi-
Head Self-Attention (MHSA) and Feed Forward Network (FFN) layers, with
residual layers in-between:

$\displaystyle\mathcal{Z}^{(l)}$
$\displaystyle=\textbf{\emph{MSHA}}(\mathcal{X}^{(l)})+\mathcal{X}^{(l)}$ (1)
$\displaystyle\mathcal{X}^{(l+1)}$
$\displaystyle=\textbf{\emph{FFN}}(\mathcal{Z}^{(l)})+\mathcal{Z}^{(l)}$
$\displaystyle=\left[X^{(l+1)}_{0},...,X^{(l+1)}_{N}\right]\in\mathbb{R}^{(N+1)\times
D}$

In the last output sequence, the regression token ${X}^{(L)}_{0}$ is used as
input to the final Multi-Layer Perceptron (MLP) for prediction (classification
or regression).

###  2.2 Surface Masked AutoEncoder

The Masked AutoEncoder (MAE) [13] frames self-supervision of vision
transformers as an image reconstruction problem. During training, a fraction
of the input signal is randomly masked out, and the model aims to reconstruct
the missing patches.

####  2.2.1 Pre-training network

Similarly to [13] the sMAE architecture follows an asymmetric encoder-decoder
design (Fig 1). The encoder processes only the unmasked tokens of the input
sequence, while the decoder receives input at the full resolution of the
original embedded sequence. This includes the latent representations of the
unmasked tokens and the masked tokens (empty embeddings). In this
implementation, the encoder is a SiT-tiny (with $L=12$ transformer encoder
layers, with 3 MHSA heads per layer), and the decoder is computationally
lighter: built from 1/4 of the number of layers of a SiT-tiny model and the
same number of MHSA heads. Finally, a linear layer is applied to project the
tokens in the final sequence back into the input patch resolution (Figure
1.e). Here, the decoder is only used for pre-training, whereas the encoder is
used for finetuning downstream tasks. Two different types of finetuning were
performed: end-to-end finetuning where all encoder layers are retrained
following SSL with the MLP head; or partial finetuning, in particular linear
probing, where all encoder layers are kept frozen, and only the MLP head is
trained.

####  2.2.2 Reconstruction Task

After adding positional embeddings $E_{pos\\_enc}$ to the input sequence
$\mathcal{X}$, a set of token indices $J_{mask}$ is randomly sampled to mask a
fraction of the input sequence (see Fig 1.b) such that $J=J_{mask}\cup
J_{unmasked}$, where $J=\\{0,1...,N\\}$ and $J_{mask}\cap
J_{unmask}=\emptyset$. The sequence of unmasked tokens
$\mathcal{X}_{unmasked}=\allowbreak\left\\{X^{0}_{i}|i\in
J_{unmasked}\right\\}$ is first encoded (Fig 1.c); then the output
$\mathcal{X}^{(L)}_{unmasked}$ (Eq.1) is concatenated to the set of masked
embeddings and the entire sequence is unshuffled, to restore the original
token order (Fig 1.d). Fixed positional embeddings
$E_{pos\\_dec}\in\mathbb{R}^{(N+1)\times D}$ are added to this sequence of
embeddings, before passing it to the decoder, which seeks to output a full-
resolution reconstruction of the image. Note, the loss (Mean Square Error,
MSE) is computed only between the masked patches, following [13].

####  2.2.3 Masked Patch Prediction

The sMAE methodology is compared to the Masked Patch Prediction (MPP) task,
used previously in [7], which in turn was adapted from [9, 8]. It employs an
autoencoder architecture that is trained to reconstruct the entire image
sequence while corrupting some of the input patches through masking or
swapping. Following [9], we corrupt 50% of the input patches randomly: by
replacing the patches with a masked (empty) token (40%), using another patch
embedding from the sequence at random (5%), or preserving their original
embeddings (5%). In contrast to sMAE, the MPP methodology optimises
reconstruction by computing the MSE loss for all patches. In this approach,
the SiT encoder processes the entire sequence of patches, and the decoder is
limited to projecting the last encoder’s sequence to the input patch
resolution.

##  3 Experiments and Results

We evaluate the performance of the sMAE framework for cortical phenotype
regression from neonatal cortical surface features from the dHCP under three
supervised training schemes: end-to-end finetuning on all data, end-to-end
finetuning on a subset of the data, and linear probing on all data.

###  3.1 Dataset and Tasks

####  3.1.1 dHCP

Data from the dHCP consists of cortical surface meshes and metrics (sulcal
depth, curvature, cortical thickness and T1w/T2w myelination) derived from T1-
and T2-weighted magnetic resonance images (MRI) [19]. We use 580 scans from
419 term neonates (born after $37$ weeks gestation) and 111 preterm neonates
(born prior to $37$ weeks gestation). 95 preterm neonates were scanned twice,
once shortly after birth, and once at term-equivalent age. Phenotype
regression was preformed on two tasks: prediction of postmenstrual age (PMA)
at scan, and gestational age (GA) at birth. For PMA prediction training data
was drawn from the scans of term-born neonates and preterm neonates’ first
scans (26.71 to 44.71 weeks PMA) with a train:val:test split of 423:53:54
examples. The GA model predicts the degree of prematurity from the
participants’ term-age scans (scans from term neonates and preterm neonate’s
second scans) with a split of 411:51:52 examples. In both cases, balanced
distribution of examples from each age bin was ensured. For the end-to-end
finetuning of PMA with partial data, three subsets of the training data were
generated with respectively 10%, 20%, and 50% of the full dHCP training
dataset. The distribution of scan age across the full training set was
preserved while generating the subsets.

Figure 2: Sulcal depth reconstructions from the surface Masked AutoEncoder
(sMAE) trained for different masking ratios (%). Below, we compare the outputs
of two sMAE trained with 50% patching with and without using positional
embeddings. sMAE networks were trained for a maximum of 100 epochs.

####  3.1.2 UK Biobank (UKB)

Data collected from 4063 subjects (1896 biological females) aged between 46
and 83 years [20, 1] was also used. In this case, cortical surfaces and
metrics (similar to dHCP) were reconstructed using the HCP structural pipeline
[12]. We split the dataset into train:val:test sets of respective size
2865:588:610.

Encoder |  Pre-training | PMA at Scan | Gestational Age
---|---|---|---
architecture |  method |  error $\pm$ std |  R2 |  error $\pm$ std |  R2
|  None |  0.87 $\pm$ 0.08 |  0.92 |  1.65 $\pm$ 0.11 |  0.57
SiT-tiny ico3 |  MPP |  0.66 $\pm$ 0.02 |  0.95 |  1.53 $\pm$ 0.07 |  0.60
|  Ours (sMAE) |  0.64 $\pm$ 0.03 |  0.96 |  1.22 $\pm$ 0.02 |  0.77
Table 1: Fine-tuning results of PMA and GA under different SSL frameworks: from scratch (None), after MPP and sMAE. Each SiT-tiny encoder is trained three times, averaged test prediction errors and stds are shown in the table. Masking Ratio |  Reconstruction Error |  Gestational Age
---|---|---
25% |  0.348 |  1.51 $\pm$ 0.10
50% |  0.290 |  1.35 $\pm$ 0.02
75% |  0.301 |  1.42 $\pm$ 0.04
90% |  0.302 |  1.44 $\pm$ 0.07
Table 2: Masking ratio selection. Reconstruction errors (MSE) on validation
set for masked patches only. For each masking ratio configuration, the sMAE
encoder was fine-tuned for GA prediction and 200 epochs. Validation prediction
errors with stds across three runs are reported.

####  3.1.3 Training Settings

Prior to training, data normalisation was performed on a per-subject and per-
channel basis. Additionally, each channel was clipped to a range of values
within $\pm 3$ standard deviations (std). All models were trained on a single
RTX 3090 NVIDIA GPU using stochastic gradient descent (SGD) optimization with
a momentum of 0.9 and a learning rate of $1e^{-4}$. For the linear probing
experiments, the learning rate was reduced to $1e^{-5}$.

###  3.2 Results

####  3.2.1 Effect of Masking Ratio

We first evaluate the effect of the masking ratio hyperparameter of the sMAE
framework. sMAE networks were trained with different masking ratios (25%, 50%,
75% and 90%) on dHCP data and evaluated for both reconstruction quality and
prediction performance in a downstream task (GA regression). Table 2 reports
the best MSE reconstruction errors and mean absolute error (referred to as
prediction error in the following) on the validation set, averaged across
three fine-tuning runs. Figure 2 shows an example of the reconstruction
quality for the validation set for each sMAE masking ratio, indicating that
sMAE models capture individual cortical features even with high masking
ratios, which suggests some capabilities of self-attention to model complex
dependencies between brain regions. Overall, the 50% masking ratio offered the
best quantitative validation and was used in all following experiments. Such
observations are aligned with those in [13], where it was noted that masking a
high percentage of patches is necessary to reduce redundancy and create a
challenging self-supervisory task which leads to learning more meaningful
weights.

####  3.2.2 Performance on end-to-end finetuning.

We benchmark the use of the sMAE against the MPP for pre-training SiT-tiny
models and compare against training models from scratch for 1000 epochs for
the tasks of PMA at scan and GA at birth regression. Results in Table 1 show
that sMAE on dHCP data improved performance in all cases compared to training
from scratch and with an MPP: 26% and 3% respectively (on PMA prediction) and
26% and 20% respectively (GA regression). In addition to performance
improvements, a major advantage of using SSL as pre-training for the SiT is
the speed gain in the process of training the supervised task. Here, faster
convergence is defined as no improvement in validation loss and prediction
error for at least 20 consecutive epochs. Figure 3 a) and b) show the
validation loss and prediction error across three fine-tuning runs of the sMAE
encoder (SiT-tiny) for the task of GA prediction. On average, sMAE pre-
training resulted in a 66% faster convergence on GA prediction and 80% on PMA,
compared to training from scratch.

Figure 3: GA prediction results comparing training from scratch, finetuning
from MPP and from sMAE. Left: validation loss; Right: validation error. Mean
and std performances (across 3 runs) are plotted. The y-axis is shown in log-
scale.

####  3.2.3 Evaluating transfer learning capabilities with sMAE.

To assess the potential of sMAE SSL for transfer learning on smaller datasets,
we first investigate the effectiveness of end-to-end fine-tuning, using
subsets of the entire dHCP dataset (10%, 20% or 50%). Results are reported for
the PMA task in Table 3.a. Remarkably, even with only 20% of the samples for
fine-tuning, the model pretrained on dHCP data achieved better results than
training from scratch using the entire dataset (Table 1). Pre-training on UKB
also benefits the end-to-end finetuning in all dHCP finetuning sizes, compared
to training from scratch.

Next, a linear probing experiment was conducted using sMAE models pre-trained
either with dHCP or UKB data (referred as sMAE (dHCP) and sMAE (UKB) in Table
3.a and 3.b). In both cases, finetuning is performed by training only the
regression head (577 parameters) while keeping all encoder layers frozen, and
demonstrates comparable performance to a full SiT-tiny (5M parameters) trained
from scratch (Table 3.a).

% dHCP data |  None |  sMAE (UKB) |  sMAE (dHCP)
---|---|---|---
10% |  1.73 |  1.58 |  0.84
20% |  1.48 |  1.36 |  0.76
50% |  1.36 |  0.92 |  0.66
100% |  0.81 |  0.70 |  0.61
(a) End-to-end finetuning on different dHCP dataset sizes from scratch (None);
or following sMAE pre-training on UKB or dHCP data.

Pre-training (dataset) |  Linear Probing
---|---
None |  1.63
sMAE (UKB) |  0.84
sMAE (dHCP) |  0.78
(b) In linear probing, only 577 parameters are trained (MLP head). Finetuning
is done on all available dHCP training data.

Table 3: Transfer Learning experiments on PMA prediction. In both tables, sMAE
(dHCP) and sMAE (UKB) models are pre-trained on 100% of available training
data. Best prediction errors on the PMA dHCP test set are reported.

##  4 Discussion

In conclusion, sMAE SSL pretraining of SiTs demonstrates improved performance
relative to MPP pretraining and training from scratch on phenotype regression
tasks, suggesting that image reconstruction with masked autoencoders learns
robust cortical feature representations that generalise well to downstream
tasks. Importantly, findings from finetuning on partial datasets, and linear
probing of UKB, suggests that sMAE pre-training could generalise the
application of SiTs to much smaller cortical imaging datasets. However, it is
worth noting some drop in performance when moving from dHCP to UKB pre-
training, which might point to issues of domain shift. Note that while data
augmentation techniques were not employed in this study (to assess gains
solely from pre-training) incorporating such techniques in future studies
could further enhance generalisation performance.

## References

  * [1] Alfaro-Almagro, F., Jenkinson, M., Bangerter, N.K., Andersson, J.L., Griffanti, L., Douaud, G., Sotiropoulos, S.N., Jbabdi, S., Hernandez-Fernandez, M., Vallee, E., et al.: Image processing and quality control for the first 10,000 brain imaging datasets from uk biobank. Neuroimage 166, 400–424 (2018)
  * [2] Azizi, S., Mustafa, B., Ryan, F., Beaver, Z., Freyberg, J., Deaton, J., Loh, A., Karthikesalingam, A., Kornblith, S., Chen, T., Natarajan, V., Norouzi, M.: Big Self-Supervised Models Advance Medical Image Classification (Apr 2021), http://arxiv.org/abs/2101.05224, arXiv:2101.05224 [cs, eess]
  * [3] Bello, G.A., Dawes, T.J.W., Duan, J., Biffi, C., de Marvao, A., Howard, L.S.G.E., Gibbs, J.S.R., Wilkins, M.R., Cook, S.A., Rueckert, D., O’Regan, D.P.: Deep-learning cardiac motion analysis for human survival prediction. Nature Machine Intelligence 1(2), 95–104 (feb 2019). https://doi.org/10.1038/s42256-019-0019-2, https://doi.org/10.1038%2Fs42256-019-0019-2
  * [4] Bronstein, M.M., Bruna, J., Cohen, T., Veličković, P.: Geometric deep learning: Grids, groups, graphs, geodesics, and gauges (2021). https://doi.org/10.48550/ARXIV.2104.13478, https://arxiv.org/abs/2104.13478
  * [5] Caron, M., Touvron, H., Misra, I., Jégou, H., Mairal, J., Bojanowski, P., Joulin, A.: Emerging properties in self-supervised vision transformers (2021). https://doi.org/10.48550/ARXIV.2104.14294, https://arxiv.org/abs/2104.14294
  * [6] Chen, L., Bentley, P., Mori, K., Misawa, K., Fujiwara, M., Rueckert, D.: Self-supervised learning for medical image analysis using image context restoration. Medical Image Analysis 58, 101539 (Dec 2019). https://doi.org/10.1016/j.media.2019.101539, https://www.sciencedirect.com/science/article/pii/S1361841518304699
  * [7] Dahan, S., Fawaz, A., Williams, L.Z., Yang, C., Coalson, T.S., Glasser, M.F., Edwards, A.D., Rueckert, D., Robinson, E.C.: Surface vision transformers: Attention-based modelling applied to cortical analysis. arXiv preprint arXiv:2203.16414 (2022)
  * [8] Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: Bert: Pre-training of deep bidirectional transformers for language understanding (2019)
  * [9] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., Houlsby, N.: An image is worth 16x16 words: Transformers for image recognition at scale. CoRR abs/2010.11929 (2020), https://arxiv.org/abs/2010.11929
  * [10] Fawaz, A., Williams, L.Z., Alansary, A., Bass, C., Gopinath, K., da Silva, M., Dahan, S., Adamson, C., Alexander, B., Thompson, D., et al.: Benchmarking geometric deep learning for cortical segmentation and neurodevelopmental phenotype prediction. bioRxiv pp. 2021–12 (2021)
  * [11] Ghesu, F.C., Georgescu, B., Mansoor, A., Yoo, Y., Neumann, D., Patel, P., Vishwanath, R.S., Balter, J.M., Cao, Y., Grbic, S., Comaniciu, D.: Self-supervised Learning from 100 Million Medical Images (Jan 2022), http://arxiv.org/abs/2201.01283, arXiv:2201.01283 [cs]
  * [12] Glasser, M., Sotiropoulos, S., Wilson, J., Coalson, T., Fischl, B., Andersson, J., Xu, J., Jbabdi, S., Webster, M., Polimeni, J., DC, V., Jenkinson, M.: The minimal preprocessing pipelines for the human connectome project. NeuroImage 80, 105 (10 2013). https://doi.org/10.1016/j.neuroimage.2013.04.127
  * [13] He, K., Chen, X., Xie, S., Li, Y., Dollár, P., Girshick, R.: Masked autoencoders are scalable vision learners (2021)
  * [14] Hou, Z., Liu, X., Cen, Y., Dong, Y., Yang, H., Wang, C., Tang, J.: Graphmae: Self-supervised masked graph autoencoders (2022)
  * [15] Kim, B.H., Ye, J.C., Kim, J.J.: Learning dynamic graph representation of brain connectome with spatio-temporal attention (2021). https://doi.org/10.48550/ARXIV.2105.13495, https://arxiv.org/abs/2105.13495
  * [16] Krishnan, R., Rajpurkar, P., Topol, E.J.: Self-supervised learning in medicine and healthcare. Nature Biomedical Engineering 6(12), 1346–1352 (Dec 2022). https://doi.org/10.1038/s41551-022-00914-1, https://www.nature.com/articles/s41551-022-00914-1, number: 12 Publisher: Nature Publishing Group
  * [17] Liang, Y., Zhao, S., Yu, B., Zhang, J., He, F.: Meshmae: Masked autoencoders for 3d mesh data analysis (2022)
  * [18] Ma, Q., Robinson, E.C., Kainz, B., Rueckert, D., Alansary, A.: Pialnn: A fast deep learning framework for cortical pial surface reconstruction (2021). https://doi.org/10.48550/ARXIV.2109.03693, https://arxiv.org/abs/2109.03693
  * [19] Makropoulos, A., Robinson, E.C., Schuh, A., Wright, R., Fitzgibbon, S., Bozek, J., Counsell, S.J., Steinweg, J., Vecchiato, K., Passerat-Palmbach, J., Lenz, G., Mortari, F., Tenev, T., Duff, E.P., Bastiani, M., Cordero-Grande, L., Hughes, E., Tusor, N., Tournier, J.D., Hutter, J., Price, A.N., Teixeira, R.P.A., Murgasova, M., Victor, S., Kelly, C., Rutherford, M.A., Smith, S.M., Edwards, A.D., Hajnal, J.V., Jenkinson, M., Rueckert, D.: The Developing Human Connectome Project: A Minimal Processing Pipeline for Neonatal Cortical Surface Reconstruction (apr 2018). https://doi.org/10.1101/125526, https://pubmed.ncbi.nlm.nih.gov/29409960/
  * [20] Miller, K.L., Alfaro-Almagro, F., Bangerter, N.K., Thomas, D.L., Yacoub, E., Xu, J., Bartsch, A.J., Jbabdi, S., Sotiropoulos, S.N., Andersson, J.L., et al.: Multimodal population brain imaging in the uk biobank prospective epidemiological study. Nature neuroscience 19(11), 1523–1536 (2016)
  * [21] Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., Sutskever, I.: Language models are unsupervised multitask learners (2019)
  * [22] Suliman, M.A., Williams, L.Z., Fawaz, A., Robinson, E.C.: A deep-discrete learning framework for spherical surface registration. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 119–129. Springer (2022)
  * [23] Xu, H., Williams, S.E., Williams, M.C., Newby, D.E., Taylor, J., Neji, R., Kunze, K.P., Niederer, S.A., Young, A.A.: Deep learning estimation of three-dimensional left atrial shape from two-chamber and four-chamber cardiac long axis views. European Heart Journal - Cardiovascular Imaging 24(5), 607–615 (02 2023). https://doi.org/10.1093/ehjci/jead010, https://doi.org/10.1093/ehjci/jead010
  * [24] Yu, X., Tang, L., Rao, Y., Huang, T., Zhou, J., Lu, J.: Point-bert: Pre-training 3d point cloud transformers with masked point modeling (2022)


10mm10mm5mm5mm

# Application of superalgebra homology groups to distinguish Engel-like
structures

Kentaro Mikami111 Akita University, Japan<EMAIL_ADDRESS>This
work is supported by JSPS KAKENHI Grant Number 22K03306.  Tadayoshi
Mizutani222Saitama University, Japan Hajime Sato 333Nagoya University, Japan

## Abstract

For a long time, tangent bundle of a manifold and the direct sum of multi-
vector fields is a prototype of $\mathbb{Z}$-graded superalgebra with the
Schouten bracket, and the grading is defined by $a-1$ for $a$-multivector
fields. Recently, we noticed the direct sum of differential forms of a
manifold has $\mathbb{Z}$-graded superalgebra structure with a super bracket
$[A,B]=(-1)^{a}d(A\wedge B)$ for $a$-form $A$, and the grading of $a$-form is
given by $-a-1$. By Lie derivation, we also see that
$\sum\limits_{p=0}^{n}\Lambda^{p}\mathrm{T}^{*}(M)\oplus\mathrm{T}(M)$ has
$\mathbb{Z}$-graded Lie superalgebra structure, which is a “super”
superalgebra of the two superalgebra described above.

For a given $\mathbb{Z}$-graded Lie superalgebra, there is a notion of the
weighted (co)chain complex and (co)homology groups. In general, those objects
are infinite dimensional and hard to understand the entire properties. If we
restrict the manifold above to a finite dimensional Lie group, and multivector
fields and differential forms to (left) invariant fields and forms, then
(co)chain spaces become finite dimensional. Still studying $\mathbb{Z}$-graded
Lie super algebra, which is a generalization of Lie algebra, we encounter
complicated manipulation of sign changes depending on each degree. In this
note, we introduce our trial of using Maple software to gather possible
equations of “Engel like” Lie algebra structures by Jacobi identify, then
simply solve them and get six possibilities. Getting weighted chain complex
and manipulate its homology groups, we claim that those six possibilities are
not isomorphic as Lie algebras in generic.

## Keywords

$\mathbb{Z}$-graded Lie super algebra (which is a generalization of Lie
algebra), super bracket, the Schouten bracket, chain spaces, the boundary
operator, the homology groups, the Betti numbers, and the Euler number.

##  1 Introduction

For a 4-dimensional manifold $M$, if rank 2 distribution $D$ satisfies
$D[2]:=D+[D,D]$ has rank 3 and $D[3]:=D[2]+[D[2],D[2]]$ has rank 4, then the
pair $(M,D)$ is called an Engel structure. Here, we apply the above notion for
4-dimensional Lie algebra $\mathfrak{g}$. The pair $(\mathfrak{g},D)$ is a
Engel-type if $D$ is a 2-dimensional subspace of $\mathfrak{g}$, and
$\textrm{dim}D[2]=3$ where $D[2]:=D+[D,D]$, and $\textrm{dim}D[3]=4$ where
$D[3]:=D[2]+[D[2],D[2]]$. From the dimensional restriction, we see a specific
basis $y_{1},y_{2}$ so that $[y_{1},y_{2}]=y_{3}\in[D,D]$. We choose $y_{4}$
by $[y_{1},y_{3}]=y_{4}$. Thus,

$\displaystyle[y_{1},y_{2}]$ $\displaystyle=y_{3}\;,\qquad$
$\displaystyle[y_{2},y_{1}]$ $\displaystyle=-y_{3}$
$\displaystyle\qquad[y_{1},y_{3}]$ $\displaystyle=y_{4}\;,\quad$
$\displaystyle[y_{3},y_{1}]$ $\displaystyle=-y_{4}$
$\displaystyle[y_{1},y_{4}]$ $\displaystyle=\sum_{i}C_{1,4,i}y_{i}\;,\quad$
$\displaystyle[y_{4},y_{1}]$ $\displaystyle=-\sum_{i}C_{1,4,i}y_{i}$
$\displaystyle\qquad[y_{2},y_{3}]$
$\displaystyle=\sum_{i}C_{2,3,i}y_{i}\;,\quad$ $\displaystyle[y_{3},y_{2}]$
$\displaystyle=-\sum_{i}C_{2,3,i}y_{i}$ $\displaystyle[y_{2},y_{4}]$
$\displaystyle=\sum_{i}C_{2,4,i}y_{i}\;,\quad$ $\displaystyle[y_{4},y_{2}]$
$\displaystyle=-\sum_{i}C_{2,4,i}y_{i}$ $\displaystyle\qquad[y_{3},y_{4}]$
$\displaystyle=\sum_{i}C_{3,4,i}y_{i}\;,\quad$ $\displaystyle[y_{4},y_{3}]$
$\displaystyle=-\sum_{i}C_{3,4,i}y_{i}\;.$

We let know Maple2021 about the relations above and check Jacobi identities by
our maple script Engel-try-1.mpl.

⬇

(* Jun 19, 2022 dim gg = 4; D is a 2dim plane of gg.

D^{2} := D + [D, D]: D^{3} := D^{2} + [ D^{2}, D^{2} ]:

If dim D^{2} = 3 and dim D^{3} = 4, ( gg , D ) is called an Engel structure.

Now want to find concrete Lie algebra having Engel structure and whose
Engel’s. *)

C := table():

with(difforms): defform(y=1, C=0):

n := 4: ybase := [seq( y[i],i=1..n) ]: gBase := ybase:

read"Sbt-renew-v1.mpl": # SbtN and those are in BIG.mla

# with(‘km/LieAlg/Sbt-renew-v1.mpl‘):

kmread("km/LieAlg/Sbt-renew-v1.mpl"):

#For test kmread("X/Y/zz.mpl");

for i to n do SbtT[ y[i], y[i] ] := 0 od:

SbtT[y[1],y[2]] := y[3]: SbtT[y[2],y[1]] := - y[3]:

SbtT[y[1],y[3]] := y[4] : SbtT[y[3],y[1]] := - y[4] :

SbtT[y[1],y[4]] := add(C[1,4,j]*y[j] , j=1..n):

SbtT[y[4],y[1]] := -add(C[1,4,j]*y[j] , j=1..n):

myVar := NULL: myVar ,= seq( C[1,4,j], j=1..n):

for i from 2 to n do for k from i+1 to n do

SbtT[y[i],y[k]] := add( C[i,k,j]*y[j] , j=1..n) :

SbtT[y[k],y[i]] := -add( C[i,k,j]*y[j] , j=1..n) :

myVar ,= seq( C[i,k,j], j=1..n): od od:

myJac := proc( A,B,C ) local a,b,c;

SbtN(A, SbtN(B,C)) + SbtN(B, SbtN(C,A)) + SbtN(C, SbtN(A,B)) : end proc:

mySkew := proc( A,B ) local a,b,c; SbtN(A, B) + SbtN(B, A) : end proc:

(* ukeComm := NULL:

for i to n do for j to n do ukeComm ,= mySkew( y[i], y[j] ) od od: *)

ukeJ := NULL: for i to n do for j from i+1 to n do for k from j+1 to n do

ukeJ ,= myJac ( y[i], y[j], y[k] ) od od od:

myEqn := NULL:

for i to n do myEqn ,= seq( diff( ukeJ[i], y[j] ), j=1..n ) od:

mySol := solve({myEqn}, {myVar});

#NG mySolK := map( simplify, mySol ):

mySolK := map( curry( map, simplify ), [ mySol ] ):

mySolTools := SolveTools[PolynomialSystem]({myEqn}, {myVar});

mySolToolsK := map( curry(map, simplify), [mySolTools] ):

for j to nops( mySolK ) do

mySolInd,mySolDep := selectremove( i-> lhs(i)= rhs(i), mySolK[j] ):

ourSolInd[j] := map( lhs, mySolInd ):

ourSolDep[j] := mySolDep :

mySolToolInd, mySolToolDep := selectremove( i-> lhs(i)= rhs(i), mySolToolsK[j]
):

ourSolToolInd[j] := map( lhs, mySolToolInd ):

ourSolToolDep[j] := mySolToolDep:

od:

(* UKE := table(sparse=NULL): SbtTT := table():

for i to nops([mySol]) do print(i,"", mySol[i]):

for j to n do for k from j+1 to n do

UKE[i] ,= SbtTT [y[j],y[k]] = simplify( SbtT[y[j],y[k]], mySol[i] ) od od:

print(UKE[i]): od: *)

# assign( UKE[1] ): BktT[ y[i], y[j] ]:

# save mySol, SbtT, UKE, "Engel-try-1-out2.txt":

# save ourSolToolInd, ourSolToolDep, ourSolInd, ourSolDep, SbtT, "Engel-
try-1-out4.txt":

(* Important caution. ourSol and ourSotTool are not exactly equal. 1 to 4 are
equal but

5 and 6 are swapped. So, we do use one way, do not mixed using. Jul 14, 2022
*)

(* i := 1: while( X[i] <> [] ) do some job : i := i+1 od: *)

We show how Maple works well.

See pages 1 of Engel-try-out1.pdf

See pages 1 of Engel-try-out2.pdf

See pages 1 of Engel-try-out3.pdf

See pages 1 of Engel-try-out4.pdf

See pages 1 of Engel-try-out6.pdf

See pages 1 of Engel-try-out5.pdf

As you expect, in the original Engel-try-1.mpl if we declare save at the
almost end of the file, we get an output file Engel-try-1-out4.txt. Preparing
the general form of Lie bracket candidate of Engel-type, we gather Jacobi
conditions myEqn. Solving them, we have 6 cases, and denote them as
ourSolInd[i] and ourSolDep[i] for $i=1,\ldots,6$. As you see, the independent
variable $C_{1,4,4}$ of the second and third cases is assumed to be non-zero.
ourSolDep[2] looks rather complicated comparing the others. Our main purpose
of this note is to distinguish those 6 cases are Lie algebra isomorphic or not
by watching their homology groups in superalgebra sense. Our strategy is
simple, that is, if two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ are
isomorphic as Lie algebras, their superalgebras are isomorphic, and so
weighted homology groups coincide for each primary weight and the superalgebra
type. Contrary, if the corresponding homology groups are not equal for some
same type of superalgebras and for some primary weight, then the original Lie
algebras are not isomorphic.

##  2 Application of super homology of tangent type
$\displaystyle\mathop{\oplus}_{\ell=1}^{4}\Lambda^{\ell}\mathfrak{g}$

We have six types after solving the Jacobi identity conditions. In the type 2
and 3, some coefficients are fractions of denominator $C_{1,4,4}$, this means
it is non-zero. Looking at those six types, the type 2 seems to be more
complicated.

We have a super homology group theory for superalgebra, especially derived
from Lie algebras which come from Engel like structure as we got above. We ask
Maple to manipulate superalgebra homology groups with the primary weight 0, 1,
or 2. In the following, we show the outputs by Maple for the six types in
”generic”. In the weight 0 case, the super homology group is just the Lie
algebra homology group and it is natural to treat 0-th chain space
$\mathfrak{g}^{0}=\mathbb{R}$, and modify the output by Maple and we conclude
the Euler number is 0 for the weight 0 case, too.

Weight 0, Common Headers | Weight 1, Common Headers | Weight 0, Common Headers
---|---|---
$\left[\begin{array}[]{*{6}{c}}\text{m}&0&1&2&3&4\\\ \text{SpaD}&1&4&6&4&1\\\ \end{array}\right]$ |  $\left[\begin{array}[]{*{6}{c}}\text{m}&1&2&3&4&5\\\ \text{SpaD}&6&24&36&24&6\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{m}&1&2&3&4&5&6\\\ \text{SpaD}&4&37&108&142&88&21\\\ \end{array}\right]$
Weight 0 of Engel ourType 1 | Weight 1 of Engel ourType 1 | Weight 2 of Engel ourType 1
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&1&4&4&1&0\\\ \text{Bett}&1&2&1&0&0\\\ \end{array}\right]$ |  $\left[\begin{array}[]{*{6}{c}}\text{KerD}&6&19&19&6&0\\\ \text{Bett}&1&2&1&0&0\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&33&76&68&21&0\\\ \text{Bett}&0&1&2&1&0&0\\\ \end{array}\right]$
Weight 0 of Engel ourType 2 | Weight 1 of Engel ourType 2 | Weight 2 of Engel ourType 2
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&1&4&3&1&0\\\ \text{Bett}&1&1&0&0&0\\\ \end{array}\right]$ |  $\left[\begin{array}[]{*{6}{c}}\text{KerD}&6&18&18&6&0\\\ \text{Bett}&0&0&0&0&0\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&33&75&67&21&0\\\ \text{Bett}&0&0&0&0&0&0\\\ \end{array}\right]$
Weight 0 of Engel ourType 3 | Weight 1 of Engel ourType 3 | Weight 2 of Engel ourType 3
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&1&4&3&1&0\\\ \text{Bett}&1&1&0&0&0\\\ \end{array}\right]$ |  $\left[\begin{array}[]{*{6}{c}}\text{KerD}&6&18&18&6&0\\\ \text{Bett}&0&0&0&0&0\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&33&75&67&21&0\\\ \text{Bett}&0&0&0&0&0&0\\\ \end{array}\right]$
Weight 0 of Engel ourType 4 | Weight 1 of Engel ourType 4 | Weight 2 of Engel ourType 4
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&1&4&3&1&0\\\ \text{Bett}&1&1&0&0&0\\\ \end{array}\right]$ |  $\left[\begin{array}[]{*{6}{c}}\text{KerD}&6&18&18&6&0\\\ \text{Bett}&0&0&0&0&0\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&33&75&67&21&0\\\ \text{Bett}&0&0&0&0&0&0\\\ \end{array}\right]$
Weight 0 of Engel ourType 5 | Weight 1 of Engel ourType 5 | Weight 2 of Engel ourType 5
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&1&4&3&1&1\\\ \text{Bett}&1&1&0&1&1\\\ \end{array}\right]$ |  $\left[\begin{array}[]{*{6}{c}}\text{KerD}&6&18&18&6&0\\\ \text{Bett}&0&0&0&0&0\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&33&77&67&23&1\\\ \text{Bett}&0&2&2&2&3&1\\\ \end{array}\right]$
Weight 0 of Engel ourType 6 | Weight 1 of Engel ourType 6 | Weight 2 of Engel ourType 6
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&1&4&3&1&1\\\ \text{Bett}&1&1&0&1&1\\\ \end{array}\right]$ |  $\left[\begin{array}[]{*{6}{c}}\text{KerD}&6&18&18&6&0\\\ \text{Bett}&0&0&0&0&0\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&33&77&67&21&2\\\ \text{Bett}&0&2&2&0&2&2\\\ \end{array}\right]$
Table 1: by Engel-mySols-v2.mpl

Then we have the proposition below.

######  Proposition 2.1.

By the three weighted generic results about Betti numbers for each type
Type(i) where $i=1\ldots 6$, those six types are divided into 4 classes
Type(1), {Type(2), Type(3), Type(4)}, Type(4) and Type(6), so that each two
Lie algebras are not isomorphic if they belong different classes.

######  Remark 2.1.

We expect to find more refinement if we try the third or more weight cases.
The word “generic” in the theorem above means we just follow Maple formal
manipulations. When we want the kernel dimension, we have two ways: one is
solve the linear equations, and the number of free parameters means the kernel
dimension. The other is to fix the Groebner basis whose number shows the rank.
For instance, assume we have a linear system $\\{ax+by=0\\}$ for $x,y$ are
independent variables and $a,b$ are general constants. Maple solve-command
shows $x=-\frac{by}{a},y=y$ and Maple Groebner:-Basis({a x + b y}, tdeg( x,y
)) shows $[ax+by]$ formally. We say those answers are “generic”. If $a\neq 0$
or $b\neq 0$, then the result is included in the generic solution. But, if
$a=b=0$, then the kernel space is 2-dimensional and the Groebner basis is
0-dimensional. We may be careful Maple does not care about this case.

##  3 Application of super homology of cotangent type
$\displaystyle\mathop{\oplus}_{\ell=0}^{4}\Lambda^{\ell}\mathfrak{g}^{*}$

We have superalgebras on the exterior algebra of differential forms of
manifold . We restrict our attention to the invariant forms of Lie groups, and
have the super homology groups. We refer to [8] about precise definitions, but
in this note, the grading for forms is sharp, namely, the grade of $p$-form is
$-(p+1)$. On the contrary, the grade of $p$-multi vector field is $p-1$. Given
a (primary) weight $wt$, if $wt$ is non-negative, then the chain space is
empty. Otherwise we have chain space bounded to $|wt|$, which is
1-dimensional, consists of
$\displaystyle\underbrace{1\bigtriangleup\cdots\bigtriangleup 1}_{|wt|}$,
which is a kernel element.

$\begin{array}[]{|c | *{5}{r}|}\hline\cr m&1&2&3&4&5\\\ \hline\cr\text{SpaceDim}&1&28&12&4&1\\\ \text{KerDim}&1&&&&1\\\ \text{Betti}&&&&&1\\\ \hline\cr\end{array}$

We concentrate for the weight $-5$ here. We follow the definition of $m$ -th
chain space. It consists of
$\displaystyle\mathfrak{h}_{-1}^{a_{1}}\wedge\mathfrak{h}_{-2}^{a_{2}}\wedge\mathfrak{h}_{-3}^{a_{3}}\wedge\mathfrak{h}_{-4}^{a_{4}}\wedge\mathfrak{h}_{-5}^{a_{5}}$
where $\displaystyle a_{1}+\cdots+a_{5}=m$ and
$\displaystyle(-1)a_{1}+\cdots+(-5)a_{5}=-5$. Directly, we see that $m<=5$ and
$a_{i}$ comes from the Young diagrams of area 5 and length $m$. Thus the
common table becomes as seen on the right end. When $m=5$, the chain space is
$\mathfrak{h}_{-1}^{5}$ and 1-dimensional. When $m=4$, the chain space is
$\mathfrak{h}_{-1}^{3}\bigtriangleup\mathfrak{h}_{-2}$ and 4-dimensional. When
$m=3$, the chain space is
$(\mathfrak{h}_{-1}\bigtriangleup\mathfrak{h}_{-2}^{2})\oplus(\mathfrak{h}_{-1}^{2}\bigtriangleup\mathfrak{h}_{-3})$
and 12-dimensional. When $m=2$, the chain space is
$(\mathfrak{h}_{-2}\bigtriangleup\mathfrak{h}_{-3})\oplus(\mathfrak{h}_{-1}\bigtriangleup\mathfrak{h}_{-4})$
and 28-dimensional. When $m=1$, the chain space is
$\mathfrak{h}_{-5}=\verb|&^|(z_{1},z_{2},z_{3},z_{4})$ and 1-dimensional. By
manipulating weight $(-5)$ super homology groups below, we classify 6 types
into 3 as Type1, Type2 – Type4, and Type5 – Type 6, which is not new in
computation of the previous section.

See pages - of DualEngel-wt5k0.pdf

See pages - of DualEngel-wt5kk1.pdf

See pages - of DualEngel-wt5kk2.pdf

See pages - of DualEngel-wt5kk3.pdf

See pages - of DualEngel-wt5kk4.pdf

See pages - of DualEngel-wt5kk5.pdf

See pages - of DualEngel-wt5kk6.pdf

######  Remark 3.1.

Here, we explain about “generic” which was stated in the theorem 2.1 before,
by outputs in this section more precisely. In general, we follow both simple
solve and Groebner:-Basis. An easy checkpoint is denominator or possibility of
enumeration of elements of Groebner:-Basis.

  *

  * Type(1): The case of $m=2$:

$\displaystyle SOL$
$\displaystyle=q_{1}=q_{1},\ldots,q_{11}=q_{11},q_{13}=q_{13},\ldots,q_{28}=q_{28};$
$\displaystyle
q_{12}=-(C_{2,3,4}q_{6}-C_{2,3,4}q_{15}+C_{2,3,4}q_{20}-C_{2,3,4}q_{27}-q_{17}+q_{22}-q_{28}$
$\displaystyle\qquad\quad-\frac{2}{C_{1,4,4}}(-C_{2,4,4}q_{6}+C_{2,4,4}q_{15}-C_{2,4,4}q_{20}+C_{2,4,4}q_{27}))$
$\displaystyle EGB$
$\displaystyle=[q_{6}(C_{1,4,4}C_{2,3,4}-2C_{2,4,4})+q_{12}C_{1,4,4}+(-C_{1,4,4}C_{2,3,4}+2C_{2,4,4})q_{15}-q_{17}C_{1,4,4}+q_{20}(C_{1,4,4}C_{2,3,4}-2C_{2,4,4})$
$\displaystyle+q_{22}C_{1,4,4}+(-C_{1,4,4}C_{2,3,4}+2C_{2,4,4})q_{27}-q_{28}C_{1,4,4}]:$

$C_{1,4,4}\neq 0$ is in generic. If $C_{1,4,4}=0$ then ORG = EGB =
[$-2C_{2,4,4}(q_{6}-q_{15}+q_{20}-q_{27})$ ], and so if $C_{2,4,4}\neq 0$ then
the case is in generic. If $C_{1,4,4}=0$ and $C_{2,4,4}=0$, then rank is 0 and
the kernel dim is 28.

  * Type(1): The case of $m=3$:

$\displaystyle SOL$
$\displaystyle=\\{q_{1}=q_{1},q_{2}=0,q_{3}=0,q_{4}=0,q_{5}=0,q_{6}=0,q_{7}=q_{7},q_{8}=q_{8},q_{9}=q_{9},$
$\displaystyle\qquad
q_{10}=(C_{1,4,4}^{2}C_{2,3,4}q_{8}+C_{1,4,3}C_{2,3,4}q_{8}-C_{1,4,4}C_{2,3,4}q_{9}-C_{1,4,4}C_{2,4,4}q_{8}+C_{2,4,4}q_{9})/C_{1,4,3},$
$\displaystyle\qquad
q_{11}=-C_{1,4,4}C_{2,3,4}q_{8}+C_{2,3,4}q_{9}+C_{2,4,4}q_{8},q_{12}=0\\};$
$\displaystyle EGB$
$\displaystyle=[q_{12},q_{9}(C_{1,4,3}C_{2,3,4}^{2}+C_{1,4,4}C_{2,3,4}C_{2,4,4}-C_{2,4,4}^{2})+q_{10}(-C_{1,4,3}C_{1,4,4}C_{2,3,4}+C_{1,4,3}C_{2,4,4})$
$\displaystyle\qquad\quad+q_{11}(-C_{1,4,4}^{2}C_{2,3,4}-C_{1,4,3}C_{2,3,4}+C_{1,4,4}C_{2,4,4}),$
$\displaystyle(C_{1,4,3}C_{2,3,4}^{2}+C_{1,4,4}C_{2,3,4}C_{2,4,4}-C_{2,4,4}^{2})q_{8}-C_{1,4,3}C_{2,3,4}q_{10}+(-C_{1,4,4}C_{2,3,4}+C_{2,4,4})q_{11},q_{6},q_{5},q_{4},q_{3},q_{2}];$

Thus, if $C_{1,4,3}\neq 0$ then it is generic and rank is 8. If $C_{1,4,3}=0$,
then $EGB=[q_{12},A,B,q_{6},q_{5},q_{4},q_{3},q_{2}]$ where

$\displaystyle A$
$\displaystyle=(C_{1,4,4}C_{2,3,4}-C_{2,4,4})(C_{2,4,4}q_{9}-C_{1,4,4}q_{11})\;,$
and $\displaystyle B$
$\displaystyle=(C_{1,4,4}C_{2,3,4}-C_{2,4,4})(C_{2,4,4}q_{8}-q_{11})\;.$

If $C_{1,4,4}C_{2,3,4}-C_{2,4,4}=0$, $A=B=0$ and the rank is 6. If
$C_{1,4,4}C_{2,3,4}-C_{2,4,4}\neq 0$, then $B\neq 0$ and
$A^{\prime}=C_{2,4,4}r_{9}-C_{1,4,4}r_{11}$
$B^{\prime}=C_{2,4,4}r_{8}-r_{11}$. The wedge product
$\&{}\hat{(}A^{\prime},B^{\prime})=-C_{2,4,4}(C_{2,4,4}\&{}\hat{(}r_{8},r_{9})-C_{1,4,4}\&{}\hat{(}r_{8},r_{10})+\&{}\hat{(}r_{9},r_{10})$.
Thus, if $C_{2,4,4}=0$ then $A=C_{1,4,4}B$ and the rank is 7. If
$C_{2,4,4}\neq 0$ then $A$ and $B$ are linearly independent and the rank is 8
(in generic).

  * Type(1): Other cases are in generic.

By the same discussion, we have checked the types 2, 3, 5 and 6 are only
generic. For the types 2 and 3, $C_{1,4,4}\neq 0$ is assumed.

  *

  * Type(4): The case of $m=2$:

The kernel condition is $[-2C_{2,4,4}(q_{6}-q_{15}+q_{20}-q_{27})]$. So if
$C_{2,4,4}\neq 0$ then the kernel dimension is 27 and the rank is 1 (in
generic). If $C_{2,4,4}=0$ then the kernel dimension is 28 and the rank is 0.

  * Type(4): Other cases are in generic.

Weight 6, Common Headers | Weight 7, Common Headers
---|---
$\left[\begin{array}[]{*{6}{c}}\text{m}&2&3&4&5&6\\\ \text{SpaD}&38&32&12&4&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{m}&2&3&4&5&6&7\\\ \text{SpaD}&28&76&32&12&4&1\\\ \end{array}\right]$
Weight 6 of Engel ourType 1 | Weight 7 of Engel ourType 1
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&38&12&4&2&1\\\ \text{Bett}&18&4&2&2&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&28&50&10&4&2&1\\\ \text{Bett}&4&28&2&2&2&1\\\ \end{array}\right]$
Weight 6 of Engel ourType 2 | Weight 7 of Engel ourType 2
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&38&10&3&1&1\\\ \text{Bett}&16&1&0&1&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&28&50&9&3&1&1\\\ \text{Bett}&4&27&0&0&1&1\\\ \end{array}\right]$
Weight 6 of Engel ourType 3 | Weight 7 of Engel ourType 3
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&38&12&3&1&1\\\ \text{Bett}&18&3&0&1&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&28&52&9&3&1&1\\\ \text{Bett}&6&29&0&0&1&1\\\ \end{array}\right]$
Weight 6 of Engel ourType 4 | Weight 7 of Engel ourType 4
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&38&10&3&1&1\\\ \text{Bett}&16&1&0&1&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&28&50&9&3&1&1\\\ \text{Bett}&4&27&0&0&1&1\\\ \end{array}\right]$
Weight 6 of Engel ourType 5 | Weight 7 of Engel ourType 5
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&6&13&3&1&1\\\ \text{Bett}&19&4&0&1&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&28&53&10&3&1&1\\\ \text{Bett}&7&31&1&0&1&1\\\ \end{array}\right]$
Weight 6 of Engel ourType 6 | Weight 7 of Engel ourType 6
$\left[\begin{array}[]{*{6}{c}}\text{KerD}&38&11&3&1&1\\\ \text{Bett}&17&2&0&1&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{7}{c}}\text{KerD}&28&53&10&3&1&1\\\ \text{Bett}&7&31&1&0&1&1\\\ \end{array}\right]$
Table 2: by DualAlg-Engel-v44.mpl

##  4 Cases for the extended superalgebra
$\displaystyle\mathop{\oplus}_{\ell=0}^{4}\Lambda^{\ell}\mathfrak{g}^{*}\oplus\mathfrak{g}$

In [8], we know that
$\displaystyle\mathfrak{g}\oplus\mathop{\oplus}_{\ell=0}^{4}\Lambda^{\ell}\mathfrak{g}^{*}$
is a super superalgebra of
$\displaystyle\mathop{\oplus}_{\ell=0}^{4}\Lambda^{\ell}\mathfrak{g}^{*}$ by
using Lie derivation, in more general context. In this section, we try and see
the homology groups of those extended superalgebras for each Engel-like Lie
algebra $\mathfrak{g}$. The following output by Maple2021, which are drove by
ZeroPlus-Engel-v3.mpl and are shown below, implies that six types are divided
into 5 classes. Right now, type 2 and type 4 have the same table and are not
distinguished on this job. However, using other enhanced superalgebras with
the weight $-3$ in this section, we conclude those six types are not
isomorphic as Lie algebras. Thus, we claim the next theorem.

######  Theorem 4.1.

The six types, which come from Lie algebras axioms, are mutually not
isomorphic in generic.

######  Remark 4.1.

When the weight is $-2$, the output on the left hand side below says nothing
about Type 2 and Type 4. The weight $-3$ output on the right hand side claim
that six types are mutually not isomorphic as Lie algebras.

Weight 2, Common Headers | Weight 3, Common Headers
---|---
$\left[\begin{array}[]{*{7}{c}}\text{m}&1&2&3&4&5&6\\\ \text{SpaD}&4&17&28&22&8&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{8}{c}}\text{m}&1&2&3&4&5&6&7\\\ \text{SpaD}&6&28&53&52&28&8&1\\\ \end{array}\right]$
Weight 2 of Engel ourType 1 | Weight 3 of Engel ourType 1
$\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&13&16&10&4&1\\\ \text{Bett}&0&1&4&6&4&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{8}{c}}\text{KerD}&6&26&33&28&16&6&1\\\ \text{Bett}&4&6&9&16&14&6&1\\\ \end{array}\right]$
Weight 2 of Engel ourType 2 | Weight 3 of Engel ourType 2
$\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&13&17&10&4&1\\\ \text{Bett}&0&2&5&6&4&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{8}{c}}\text{KerD}&6&25&28&25&13&5&1\\\ \text{Bett}&3&0&1&10&10&5&1\\\ \end{array}\right]$
Weight 2 of Engel ourType 3 | Weight 3 of Engel ourType 3
$\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&13&18&10&4&1\\\ \text{Bett}&0&3&6&6&4&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{8}{c}}\text{KerD}&6&25&30&25&13&5&1\\\ \text{Bett}&3&2&3&10&10&5&1\\\ \end{array}\right]$
Weight 2 of Engel ourType 4 | Weight 3 of Engel ourType 4
$\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&13&17&10&4&1\\\ \text{Bett}&0&2&5&6&4&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{8}{c}}\text{KerD}&6&25&29&25&13&5&1\\\ \text{Bett}&3&1&2&10&10&5&1\\\ \end{array}\right]$
Weight 2 of Engel ourType 5 | Weight 3 of Engel ourType 5
$\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&13&18&10&5&1\\\ \text{Bett}&0&3&6&7&5&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{8}{c}}\text{KerD}&6&25&30&25&16&5&1\\\ \text{Bett}&3&2&3&13&13&5&1\\\ \end{array}\right]$
Weight 2 of Engel ourType 6 | Weight 3 of Engel ourType 6
$\left[\begin{array}[]{*{7}{c}}\text{KerD}&4&14&16&10&5&1\\\ \text{Bett}&1&2&4&7&5&1\\\ \end{array}\right]$ | $\left[\begin{array}[]{*{8}{c}}\text{KerD}&6&25&29&25&16&5&1\\\ \text{Bett}&3&1&2&13&13&5&1\\\ \end{array}\right]$
Table 3: by ZeroPlus-Engel-v3.mpl

##  5 Characteristic foliations

There is a notion of characteristic foliations in Engel theory, which is rank
1 distribution $\mathfrak{L}$ of Engel distribution $D$ satisfying
$[\mathfrak{L},D^{2}]\subset D^{2}$ where $D^{2}:=D+[D,D]$ (, and
$D^{3}:=D^{2}+[D^{2},D^{2}]$). In our six cases, they are given by
$ay_{1}+by_{2}$ for mySol[3], and $C_{2,3,4}y_{1}-y_{2}$ for the others, up to
constant. It will be interesting to study their contributions to super
homology discussion.

##  6 Another approach of finding Engel-like structures

So far, we studied Engel-like structures first fix dimensional conditions and
next check Lie algebra structure. As [4] points out, there is a complete
classification list of 4-dimensional Lie algebra by [10]. Since we
unfortunately do not have access to the original paper [10], we use the list
in [4].

$\displaystyle\text{Type}[1]$
$\displaystyle=\\{[y_{2},y_{4}]=y_{1},[y_{3},y_{4}]=y_{2}\\};\quad\text{Type}[2]=\\{[y_{1},y_{4}]=ay_{1},[y_{2},y_{4}]=y_{2},[y_{3},y_{4}]=y_{2}+y_{3}\\};$
$\displaystyle\text{Type}[3]$
$\displaystyle=\\{[y_{1},y_{4}]=y_{1},[y_{3},y_{4}]=y_{2}\\};\quad\text{Type}[4]=\\{[y_{1},y_{4}]=y_{1},[y_{2},y_{4}]=y_{1}+y_{2},[y_{3},y_{4}]=y_{2}+y_{3}\\};$
$\displaystyle\text{Type}[5]$
$\displaystyle=\\{[y_{1},y_{4}]=y_{1},[y_{2},y_{4}]=ay_{2},[y_{3},y_{4}]=by_{3},\text{
where }ab\neq 0\\};$ $\displaystyle\text{Type}[6]$
$\displaystyle=\\{[y_{1},y_{4}]=ay_{1},[y_{2},y_{4}]=by_{2}-y_{3},[y_{3},y_{4}]=y_{2}+by_{3},\text{
where }a\neq 0\text{ and }b\geq 0\\};$ $\displaystyle\text{Type}[7]$
$\displaystyle=\\{[y_{1},y_{4}]=2y_{1},[y_{2},y_{3}]=y_{1},[y_{2},y_{4}]=y_{2},[y_{3},y_{4}]=y_{2}+y_{3}\\};$
$\displaystyle\text{Type}[8]$
$\displaystyle=\\{[y_{2},y_{3}]=y_{1},[y_{2},y_{4}]=y_{2},[y_{3},y_{4}]=-y_{3}\\};$
$\displaystyle\text{Type}[9]$
$\displaystyle=\\{[y_{1},y_{4}]=(1+b)y_{1},[y_{2},y_{3}]=y_{1},[y_{2},y_{4}]=y_{2},[y_{3},y_{4}]=by_{3},\text{
where }-1<b\leqq 1\\};$ $\displaystyle\text{Type}[10]$
$\displaystyle=\\{[y_{2},y_{3}]=y_{1},[y_{2},y_{4}]=-y_{3},[y_{3},y_{4}]=y_{2}\\};$
$\displaystyle\text{Type}[11]$
$\displaystyle=\\{[y_{1},y_{4}]=2ay_{1},[y_{2},y_{3}]=y_{1},[y_{2},y_{4}]=ay_{2}-y_{3},[y_{3},y_{4}]=y_{2}+ay_{3}\\};$
$\displaystyle\text{Type}[12]$
$\displaystyle=\\{[y_{1},y_{3}]=y_{1},[y_{1},y_{4}]=-y_{2},[y_{2},y_{3}]=y_{2},[y_{2},y_{4}]=y_{1}\\};$

In this section, we first fix each 4-dimensional Lie algebra and try to find
2-dimensional subspace $D$ satisfying Engel-like structure. We prepare a small
proposition.

######  Proposition 6.1.

Let $\mathfrak{g}$ be a 4-dimensional Lie algebra with bracket relations by a
basis $(y_{1},\ldots,y_{4})$.

Take a 2-dimensional plane $D$ where $w_{1}=\sum_{i=1}^{4}{p}_{i}y_{i}$ and
$w_{2}=\sum_{i=1}^{4}{q}_{i}y_{i}$ are its basis. Put $w_{3}=[w_{1},w_{2}]$
and $w_{4}=[w_{1},w_{3}]$.

If $\\{w_{1},w_{2},w_{3}\\}$ and $\\{w_{1},w_{2},w_{3},w_{4}\\}$ are linearly
independent, i.e., $w_{1}\wedge w_{2}\wedge w_{3}\neq 0$ and $w_{1}\wedge
w_{2}\wedge w_{3}\wedge w_{4}\neq 0$ for some
$\\{{p}_{i}\\}_{i=1}^{4},\\{{q}_{i}\\}_{i=1}^{4}$, then $(\mathfrak{g},D)$ is
an Engel-like structure.

In this note, we call the scalar $(w_{1}\wedge w_{2}\wedge w_{3}\wedge
w_{4})/(y_{1}\wedge y_{2}\wedge y_{3}\wedge y_{4})$ above by the _E-l-C_
(Engel-like coefficient). Using the notation
$\text{Det}(i,j)={p}_{i}{q}_{j}-{p}_{j}{q}_{i}=\begin{vmatrix}{p}_{i}&{p}_{j}\\\
{q}_{i}&{q}_{j}\\\ \end{vmatrix}$, the Engel-like coefficient of Lie algebras
Type[i] (i=1..12) are given as follows.

$\displaystyle\text{E-l-C of }\text{Type}[1]$
$\displaystyle={p}_{4}\text{Det}(3,4)^{3}\qquad\qquad\qquad\qquad\text{E-l-C
of }\text{Type}[2]=(a-1)^{2}{p}_{4}\text{Det}(1,4)\text{Det}(3,4)^{2}$
$\displaystyle\text{E-l-C of }\text{Type}[3]$
$\displaystyle={p}_{4}\text{Det}(1,4)\text{Det}(3,4)^{2}\hskip
48.36967pt\text{E-l-C of }\text{Type}[4]={p}_{4}\text{Det}(3,4)^{3}$
$\displaystyle\text{E-l-C of }\text{Type}[5]$
$\displaystyle=(a-1)(b-1)(a-b){p}_{4}\text{Det}(1,4)\text{Det}(2,4)\text{Det}(3,4)$
$\displaystyle\text{E-l-C of }\text{Type}[6]$
$\displaystyle=((a-b)^{2}+1){p}_{4}\text{Det}(1,4)(\text{Det}(2,4)^{2}+\text{Det}(3,4)^{2})$
$\displaystyle\text{E-l-C of }\text{Type}[7]$
$\displaystyle=\text{Det}(3,4)^{2}({p}_{4}\text{Det}(1,4)+{p}_{4}\text{Det}(2,3)+{p}_{3}\text{Det}(3,4))$
$\displaystyle\text{E-l-C of }\text{Type}[8]$
$\displaystyle=-2\text{Det}(2,4)\text{Det}(3,4)({p}_{4}\text{Det}(1,4)-{p}_{3}\text{Det}(2,4)-{p}_{2}\text{Det}(3,4))$
$\displaystyle\text{E-l-C of }\text{Type}[9]$
$\displaystyle=-(b-1)\text{Det}(2,4)\text{Det}(3,4)\left({p}_{3}\text{Det}(1,4)+b({p}_{4}\text{Det}(1,4)-{p}_{2}\text{Det}(3,4))\right)$
$\displaystyle\text{E-l-C of}\text{Type}[10]$
$\displaystyle=(\text{Det}(2,4)^{2}+\text{Det}(3,4)^{2})({p}_{4}\text{Det}(1,4)+{p}_{2}\text{Det}(2,4)+{p}_{3}\text{Det}(3,4))$
$\displaystyle\text{E-l-C of }\text{Type}[11]$
$\displaystyle=(\text{Det}(2,4)^{2}+\text{Det}(3,4)^{2})(a^{2}{p}_{4}\text{Det}(1,4)+a{p}_{4}\text{Det}(2,3)+{p}_{4}\text{Det}(1,4)+{p}_{2}\text{Det}(2,4)+{p}_{3}\text{Det}(3,4))$
$\displaystyle\text{E-l-C of }\text{Type}[12]$
$\displaystyle={p}_{4}\text{Det}(3,4)\left(\text{Det}(1,3)^{2}+\text{Det}(1,4)^{2}+\text{Det}(2,3)^{2}+\text{Det}(2,4)^{2}+2\text{Det}(1,2)\text{Det}(3,4)\right)$

In the case of Type[1], taking $p=[0,0,0,1]$ and $q=[0,0,1,0]$, i.e.,
$w_{1}=y_{4}$ and $w_{2}=y_{3}$ give an Engel-like structure.

In the case of Type[2], if $a=1$, then there is no Engel-like structure.
Assume $a\neq 1$, then $p=[0,0,0,1]$ and $q=[1,0,1,0]$ give an Engel-like
structure.

In the case of Type[3], $p=[0,0,0,1]$ and $q=[1,0,1,0]$ give an Engel-like
structure like as the second half of Type[2].

The case Type[4] is the same with Type[1].

In the case of Type[5], if $(a-1)(b-1)(a-b)=0$, then there is no Engel-like
structure. Otherwise, $p=[0,0,0,1]$ and $q=[1,0,1,0]$ give an Engel-like
structure.

In the case of Type[6], $p=[0,0,0,1]$ and $q=[1,1,1,0]$ give an Engel-like
structure.

In the case of Type[7], $p=[0,0,1,1]$ and $q=[0,0,0,1]$ give an Engel-like
structure.

In the case of Type[8], $p=[0,0,0,1]$ and $q=[1,1,1,0]$ give an Engel-like
structure.

In the case of Type[9], if $b-1=0$ then there is no Engel-like structure.
Otherwise, $p=[1,1,1,1]$ and $q=[0,0,0,1]$ give an Engel-like structure.

In the case of Type[10], $p=[0,0,0,1]$ and $q=[1,0,1,0]$ give an Engel-like
structure.

In the case of Type[11], $p=[0,0,1,0]$ and $q=[0,0,0,1]$ give an Engel-like
structure.

In the case of Type[12], $p=[0,1,0,1]$ and $q=[0,1,1,0]$ give an Engel-like
structure.

## References

  * [1] Jiro Adachi.  Integral curves for contact and Engel structures.  In Real and complex singularities, pages 1–32. World Sci. Publ., Hackensack, NJ, 2007.
  * [2] Jiro Adachi.  Relative local equivalence of Engel structures.  Math. Z., 273(3-4):711–733, 2013.
  * [3] Rory Biggs and Claudiu C. Remsing.  On the classification of real four-dimensional Lie groups.  J. Lie Theory, 26(4):1001–1035, 2016.
  * [4] R. Ghanam and G. Thompson.  Minimal matrix representations of four-dimensional Lie algebras.  Bull. Malays. Math. Sci. Soc. (2), 36(2):343–349, 2013.
  * [5] Kentaro Mikami and Tadayoshi Mizutani.  Euler number and Betti numbers of homology groups of pre Lie superalgebra.  arXiv:1809.08028v2, December 2018.
  * [6] Kentaro Mikami and Tadayoshi Mizutani.  Euler number of homology groups of super Lie algebra.  arXiv:1809.08028v1, September 2018.
  * [7] Kentaro Mikami and Tadayoshi Mizutani.  Deformed super brackets on forms of a manifold.  arXiv:2112.05896, Dec 2021.
  * [8] Kentaro Mikami and Tadayoshi Mizutani.  Superalgebra structure on differential forms of manifold.  arXiv:2105.09738, May 2021.
  * [9] Kentaro Mikami and Tadayoshi Mizutani.  The second Betti number of doubly weighted homology groups of some pre Lie superalgebra.  Tohoku Math. J., 74(2):301–311, 2022.  arXiv:1902.09137.
  * [10] G. M. Mubarakzjanov.  Classification of real structures of Lie algebras of fifth order.  Izv. Vysš. Učebn. Zaved. Matematika, 1963(3 (34)):99–106, 1963.
  * [11] Libor Šnobl and Pavel Winternitz.  Classification and identification of Lie algebras, volume 33 of CRM Monograph Series.  American Mathematical Society, Providence, RI, 2014.

##  Appendix A Table of bracket for Engel-like Lie algebras

ourSolDep[1]

$\displaystyle[y_{1},y_{4}]$ $\displaystyle=C_{1,4,3}y_{3}+C_{1,4,4}y_{4}$
$\displaystyle\qquad[y_{2},y_{3}]$
$\displaystyle=(-C_{1,4,4}C_{2,3,4}+C_{2,4,4})y_{3}+C_{2,3,4}y_{4}$
$\displaystyle[y_{2},y_{4}]$
$\displaystyle=(C_{1,4,3}C_{2,3,4})y_{3}+C_{2,4,4}y_{4}$
$\displaystyle\qquad[y_{3},y_{4}]$ $\displaystyle=0$

ourSolDep[2]

$\displaystyle[y_{1},y_{4}]$
$\displaystyle=\frac{C_{1,4,4}^{2}+4C_{1,4,3}}{8}\left(-(C_{1,4,4}C_{2,3,4}-2C_{2,4,4})y_{1}-C_{1,4,4}y_{2}\right)+C_{1,4,3}y_{3}+C_{1,4,4}y_{4}$
$\displaystyle[y_{2},y_{3}]$
$\displaystyle=\frac{(C_{1,4,4}^{2}+4C_{1,4,3})(C_{1,4,4}C_{2,3,4}-C_{2,4,4})}{2C_{1,4,4}^{2}}(-(C_{1,4,4}C_{2,3,4}-2C_{2,4,4})y_{1}-C_{1,4,4}y_{2})$
$\displaystyle+(-C_{1,4,4}C_{2,3,4}+C_{2,4,4})y_{3}+C_{2,3,4}y_{4}$
$\displaystyle[y_{2},y_{4}]$
$\displaystyle=-\frac{1}{8}(C_{1,4,4}C_{2,3,4}-2C_{2,4,4})(C_{1,4,4}^{2}+4C_{1,4,3})C_{2,3,4}y_{1}-\frac{1}{8}(C_{1,4,4}^{2}+4C_{1,4,3})C_{1,4,4}C_{2,3,4}y_{2}$
$\displaystyle-\frac{1}{2C_{1,4,4}}(C_{1,4,4}^{3}C_{2,3,4}+2C_{1,4,3}C_{1,4,4}C_{2,3,4}-C_{1,4,4}^{2}C_{2,4,4}-4C_{1,4,3}C_{2,4,4})y_{3}+C_{2,4,4}y_{4}$
$\displaystyle[y_{3},y_{4}]$
$\displaystyle=\frac{(C_{1,4,4}^{2}+4C_{1,4,3})(C_{1,4,4}C_{2,3,4}-C_{2,4,4})}{8C_{1,4,4}^{2}}\left((C_{1,4,4}C_{2,3,4}-C_{2,4,4})(C_{1,4,4}^{2}+4C_{1,4,3})y_{1}\right.$
$\displaystyle\left.+C_{1,4,4}(C_{1,4,4}^{2}+4C_{1,4,3})y_{2}+2C_{1,4,4}^{2}y_{3}+4C_{1,4,4}y_{4}\right)$
ourSolDep[3] $\displaystyle[y_{1},y_{4}]$
$\displaystyle=-\frac{C_{1,4,2}C_{2,4,4}}{C_{1,4,4}}y_{1}+C_{1,4,2}y_{2}+C_{1,4,3}y_{3}+C_{1,4,4}y_{4}$
$\displaystyle\qquad[y_{2},y_{3}]$
$\displaystyle=\frac{C_{2,4,4}}{C_{1,4,4}}y_{4}$ $\displaystyle[y_{2},y_{4}]$
$\displaystyle=\frac{C_{2,4,4}}{C_{1,4,4}^{2}}\left(-C_{1,4,2}C_{2,4,4}y_{1}+C_{1,4,2}C_{1,4,4}y_{2}+C_{1,4,3}C_{1,4,4}y_{3}+C_{1,4,4}^{2}y_{4}\right)$
$\displaystyle\qquad[y_{3},y_{4}]$ $\displaystyle=0$ ourSolDep[4]
$\displaystyle[y_{1},y_{4}]$ $\displaystyle=0$
$\displaystyle\qquad[y_{2},y_{3}]$
$\displaystyle=C_{2,3,1}y_{1}+C_{2,4,4}y_{3}+C_{2,3,4}y_{4}$
$\displaystyle[y_{2},y_{4}]$ $\displaystyle=C_{2,4,4}y_{4}$
$\displaystyle\qquad[y_{3},y_{4}]$ $\displaystyle=0$ ourSolDep[5]
$\displaystyle[y_{1},y_{4}]$
$\displaystyle=(-C_{1,4,2}C_{2,3,4})y_{1}+C_{1,4,2}y_{2}+C_{1,4,3}y_{3}$
$\displaystyle\qquad[y_{2},y_{3}]$ $\displaystyle=C_{2,3,4}y_{4}$
$\displaystyle[y_{2},y_{4}]$
$\displaystyle=(-C_{1,4,2}C_{2,3,4}^{2})y_{1}+(C_{1,4,2}C_{2,3,4})y_{2}+(C_{1,4,3}C_{2,3,4})y_{3}$
$\displaystyle\qquad[y_{3},y_{4}]$ $\displaystyle=0$ ourSolDep[6]
$\displaystyle[y_{1},y_{4}]$ $\displaystyle=C_{1,4,3}y_{3}$
$\displaystyle\qquad[y_{2},y_{3}]$
$\displaystyle=C_{2,3,1}y_{1}+C_{3,4,4}y_{2}+C_{2,3,4}y_{4}$
$\displaystyle[y_{2},y_{4}]$
$\displaystyle=(C_{1,4,3}C_{2,3,4}+C_{3,4,4})y_{3}$
$\displaystyle\qquad[y_{3},y_{4}]$
$\displaystyle=(-C_{1,4,3}C_{2,3,1})y_{1}+(-C_{1,4,3}C_{3,4,4})y_{2}+C_{3,4,4}y_{4}$

##  Appendix B About main maple scripts in this note

Only Engel-try-1.mpl had a 2 pages full explanation, but Engel-mySols-v1.mpl,
DualAlg-Engel-v44.mpl, and ZeroPlus-Engel-v2.mpl had no chance to be explaned
at all.

In stead, we invite readers to http://math.akita-u.ac.jp/~mikami/. There we
prepare a tar-ball which includes revised maple scripts with the extension
”.mpl.with”, a small repository big.mla, and please-read-me.txt.