The beautiful physical interpretation of the decomposition of the elasticity stress tensor into its irreducible representations (mentioned in context below) has inspired the following question on representation theory.
If you interpret a $3x3$ matrix $A$ as a direct product of vectors, $A = A_jB_k$, then breaking a matrix up into a symmetric and anti-symmetric matrix $A = A_s + A_a$ is a decomposition of a 9 component tensor into a sum of a smaller 6 component and 3 component tensor to get $\bf{3\otimes 3 = 6 \oplus 3}$. Apparently I just showed the matrix representation of orthogonal group $O(3)$ is reducible.
This is as far as you can go in $O(3)$, the group of rotations in 3-space, but if you have a unit determinant you can go further and use the trace to remove some redundancy. When we can do this we call it $SO(3)$, and can further reduce the symmetric matrix into a traceless 5-component symmetric matrix and a scalar multiple of the identity (by the trace) to get a $\bf{3\otimes3 = 5\oplus 3 \oplus 1}$, i.e. $A = A_{ts} + A_t + A_a$.
I apparently just described the tensor decomposition of the $SO(3)$ orthogonal group representation into its irreducible tensor representations just there, but any orthogonal group also has a spinor representation too.
This can be given a geometric interpretation: we decomposed a rank 2 tensor into the sum of a shearing, (a symmetric rank two tensor representing a unit ellipsoid), a rank 1 tensor rotation (a vector representing a rotation) and a rank $0$ tensor (a scaling factor). (pdf page 10)
1) Is there a similar easy decomposition of a 3x3 matrix into its spinor representations?*
I mean, starting with an explicit matrix can we do some easy process to it, like taking symmetric and anti-symmetric combinations as above, and magically end up with spinor representations and show how obvious their existence is? Just so that it's not so abstract.
2) How do we give a nice geometric interpretation to the decomposition of $SO(4)$?
In other words, how do we decompose matrices in $SO(4)$ in such a way that gives a geometric interpretation to every term, illustrates why the $SO(4) \cong [SU(2) \times SU(2)]/\mathbb{Z}_2$ should be obvious, and hopefully makes the symmetries of the Riemann curvature tensor look completely obvious?
(I'm really just hoping to derive and compare $SO(3)$ spinors with $SO(4)$ spinors, since they look so different in QM vs QFT, hence two parts to this question.)