Using three Euler angles we can represent any 3-d rotation matrix.
Let's assume that allowable Euler angles must be equal to $i\dfrac{\pi}{2}$ $\,$, $i=0.. 3$ $\,$ i.e.
$\alpha = \{0,\dfrac{\pi}{2},{\pi}, \dfrac{3\pi}{2}$ (the same effect as in case of $-\dfrac{\pi}{2}$ rotation) $\}$ (such Euler angles constrain generated rotation matrices to matrices with only $-1,0,1$ entries).
and we have chosen representation XYZ for a rotation matrix (so called Tait–Bryan case) $\,$ i.e.
$R= R_x(\alpha_1)R_y(\alpha_2)R_z(\alpha_3)$
or in the more detailed form
$R= \begin{bmatrix} c_2 c_3 & - c_2 s_3 & s_2 \\ c_1 s_3 + c_3 s_1 s_2 & c_1 c_3 - s_1 s_2 s_3 & - c_2 s_1 \\ s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 + c_1 s_2 s_3 & c_1 c_2 \end{bmatrix}$
where $c_j=\cos(\alpha_j)$ and $s_j=\sin(\alpha_j)$.
Theoretically these conditions allow to get $4\times{4}\times{4}= 64$ different Euler representations for rotation matrix, but it is known (for example from How many $3 \times 3$ integer matrices are orthogonal?) that only $24$ unique rotations are possible, so for sure some Euler representations don't generate unique matrices.
- How to check (without calculating all possibilities) which of the Euler representations of rotation matrix for this kind $i\dfrac{\pi}{2}$ regular rotation generate unique representation and which don't generate such uniqueness?
- if some are not unique can we say (also without detailed calculations) how many other representations could generate the same rotation matrix as with the selected particular Euler representation?