Condition for a matrix to belong to the spin-$s$ irreducible representation of $\text{SU}(2)$ The spin-$s$ irreducible representation of $\text{su}(2)$ is generated by the three $2s+1$ dimensional spin matrices $iX,iY,iZ$. Since these spin matrices are anti-Hermitian, they span a subspace of $\text{su}(2s+1)$. It is easy to check if an arbitrary $2s+1$ dimensional anti-Hermitian matrix belongs to $\text{su}(2) \subset \text{su}(2s+1)$, as this is equivalent to solving a certain linear system.
However, using the exponential map, one can obtain the Lie group $\text{SU}(2)$ as a subgroup of $\text{SU}(2s+1)$. The elements of $\text{SU}(2)$ here are of the form
$$ U(\alpha, \beta, \gamma) = \exp(i\alpha X + i\beta Y + i\gamma Z).$$
My question is, like for the Lie algebra, is there a simple test to verify if a given element of $\text{SU}(2s + 1)$ belongs to the $\text{SU}(2)$ subgroup defined above? Ideally, I would hope that there is an algebraic condition or a special symmetry on the matrices $U(\alpha,\beta,\gamma)$ that one could readily test, but I cannot seem to find one.
 A: For lack of time, these are just some remarks which should lead you to a solution, but need some work.
First look up Section 7 of my article with Chipalkatti "The higher transvectants are redundant" at Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1671-1713.
There you will see a precise realization of the spin $s$ representation of $SU(2)$ as an action on the space of degree $2s$ homogeneous polynomials with complex coefficients in two variables. Inside the $2s+1$-dimensional vector space of such polynomials, there is a distinguished subset called the rational normal curve or Veronese variety, given by polynomials of the form $L^{2s}$ where $L$ is a linear form. Among all elements of $SU(2s+1)$, the ones coming from $SU(2)$ are the ones which keep this subset stable namely send any $L^{2s}$ to some $2s$-th power of some other linear form instead of more general polynomials. Testing membership in this subset is easy and can be done with a well known system of quadratic equations.
Now the remaining problem is now you have a characterization of the form: for every linear form in two variables $L$, the image of $L^{2s}$ by your $SU(2s+1)$ element satisfies the quadratic system. This is an infinite collection of equations because there are infnitely many $L$'s. This can be reduced to a finite system by showing you only need to test this for finitely many $L$ (Noetherian property).
