In my physics research, I stumbled upon a rather simple deformation of the $SU(2)$ algebra, and I was wondering whether it qualifies as a `$q$-deformed Lie algebra' (a notion which is unfamiliar to me). In particular, I could not find a source which was both easily digestible for a physicist as well as precise enough to clarify how rigid the notion is. I.e., I am not sure which properties to check before declaring it is (not) a $q$-deformed Lie algebra.
Consider the following three matrices, expressed in terms of the Gell-Mann matrices (i.e., generators of the $SU(3)$ Lie algebra): $$ S^x = \lambda_2, \quad S^y = \lambda_5, \quad S^z = \cos(\theta) \lambda_7 + \sin(\theta) \lambda_6. $$ Here $\theta$ is some arbitrary angle. For $\theta=0$, these three matrices generate the $SU(2)$ algebra (in particular, $[S^\alpha, S^\beta] = i \varepsilon_{\alpha\beta\gamma} S^\gamma$). However, for $\theta \neq 0$ it does not. I am wondering whether it qualifies as a $q$-deformed $SU(2)$ algebra, with $q=e^{i\theta}$. For instance, one can straightforwardly show that $$ [S^x,S^y]_q = i S^z \qquad \textrm{where we define } [A,B]_q = q AB - q^{-1} BA, $$ similarly for cyclic permutations of these matrices. Any guidance would be much appreciated. (I guess the gunshot approach is to check whether the above satisfies all the definitions of a Hopf algebra, but I am hoping for something less abstract/intense, utilizing the elementary structure of the above set-up.)
