<s>[INST] Symmetric sums and Representations of SO(3) I had tried to help someone on math.StackExchange to prove the identity:
$$ (1-Tr(A))^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4$$
I guess you could argue the left hand side is independent of basis.  Then we can diagonalize.  But I couldn't come up with an invariant way of expressing the 2nd term.  
Someone came up with $(1-\mathrm{Tr}\\,A)^2- \frac{1}{2}\mathrm{Tr}\left[(A-A^T)^2\right]=4$ and prove it.
This look a bit Pythagoras' theorem since $1 - Tr A = 2 \cos \varphi$ so the other term must be $\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2 = 2 \sin \varphi$ which I also couldn't prove.  
Is this sum related to a direct sum of representations of SO(3)?
 [/INST] Using the parametrization
$$
  A=  \begin{pmatrix} a^2+b^2-c^2-d^2&2bc-2ad &2bd+2ac \cr
 2bc+2ad &a^2-b^2+c^2-d^2&2cd-2ab \cr
 2bd-2ac &2cd+2ab &a^2-b^2-c^2+d^2\\ \end{pmatrix}, 
$$
which comes from quaternions of unit length we have
$$
(1-tr (A))^2-\frac{1}{2} tr ((A-A^t)^2)-4=
$$
$$
(9a^2 + b^2 + c^2 + d^2 + 3)(a^2 + b^2 + c^2 + d^2 - 1)=0,
$$
since $a^2+b^2+c^2+d^2=1$.
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