# Is this a valid q-deformation of $SU(2)$ via the $SU(3)$ Lie algebra?

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.)