${\mathcal{H}}= {\widetilde{\mathcal{G}}}$ a maximal subgroup of ${\widetilde{\mathcal{I}}}$, then $B(t)$ induces a continuous transformation $$\label{sigma_t_2}
\widetilde{\Sigma}_t := \left\{ \pi^{\parallel}({\boldsymbol{v}}_t) : \pi^{\perp}({\boldsymbol{v}}_t) \in \pi^{\perp}(\mathcal{V}_t(\mathbf{0})) \right\},$$ where ${\boldsymbol{v}}_t := B(t)\boldsymbol{m}$, $\boldsymbol{m} \in {\mathbb{Z}}^6$, is a point in the intermediate lattice ${\mathcal{L}}_t$, and $\mathcal{V}_t(\mathbf{0})$ denotes the Voronoi cell of ${\mathcal{L}}_t$ at the origin. The symmetry identified by ${\mathcal{G}}$ is preserved since ${\mathcal{G}}_1 \pi^{\parallel}({\boldsymbol{v}}_t) = \pi^{\parallel}({\widetilde{\mathcal{G}}}{\boldsymbol{v}}_t)$, and ${\widetilde{\mathcal{G}}}{\boldsymbol{v}}_t \in {\mathcal{L}}_t$ since ${\widetilde{\mathcal{G}}}\subseteq {\mathcal{P}}({\mathcal{L}}_t)$ for all $t \in [0,1]$. We notice that in this approach the lattice undergoes a transformation, whereas the physical and orthogonal spaces remain fixed.

As already pointed out in [@paolo], the Schur rotation and the generalised Bain strain are related. This can easily be proved with the mathematics developed so far. In particular, if $M_{{\mathcal{G}}}(t)$ is a Schur rotation associated with ${\mathcal{G}}$ as in , let us define the path $\hat{B}(t) : [0,1] \rightarrow GL(6,{\mathbb{R}})$ as $$\hat{B}(t):= M_{{\mathcal{G}}}(t)^{-1} B_0.$$ We have, using , $$\pi^{\parallel}\left(\hat{B}(t)\right) = \pi^{\parallel}\left(M_{{\mathcal{G}}}(t)^{-1} B_0\right) = \pi^{\parallel}_t(B_0),$$ and similarly for $\pi^{\perp}_t$. Therefore $$\widetilde{\Sigma}_t = \left\{ \pi^{\parallel}({\boldsymbol{v}}_t) : \pi^{\perp}({\boldsymbol{v}}_t) \in \pi^{\perp}(\mathcal{V}_t(\mathbf{0})) \right\} = \left\{ \pi^{\parallel}_t(B_0\boldsymbol{m}) : \pi^{\perp}_t(B_0\boldsymbol{m}) \in \pi^{\perp}_t(\mathcal{V}(\mathbf{0})) \right\} = \Sigma_t,$$ and moreover ${\widetilde{\mathcal{G}}}\subseteq {\mathcal{P}}({\mathcal{L}}_t)$, since ${\mathcal{P}}\left(\hat{B}(t)\right) = M_{{\mathcal{G}}}^{-1}(t) {\mathcal{P}}(B_0) M_{{\mathcal{G}}}(t)$ (this is true since, for every lattice ${\mathcal{L}}$ in 