This question may sound a bit odd, but basically I want to ask if the following algebras have any meaning in some area of math: $ A=\mathbb{C} \langle X,Y,Z \rangle /(XY+i YX,YZ+i ZY, XZ+ZX)\\ B= A / (Y^2Z-i(4X^3-XZ^2)).$ I got this algebra from considering the elliptic curve $E \leftrightarrow y^2 = 4x^3-x$, which has a non-trivial automorphism (a morphism as well as a homomorphism) of order 4 defined by $\phi: (x,y) \mapsto (-x,i y)$. Let $O(E) = \mathbb{C}[X,Y,Z]/(Y^2Z - 4X^3+XZ^2)$ be the homogenous coordinate ring of $E$ ($\mathbb{Z}$-graded) and let $\mathbb{C}(E) = \left\{ \frac{f}{g} \, \middle| \, f,g \in O(E)_i \text{ for some } i \in \mathbb{N}, g \neq 0 \right\}.$ Next, I took the twisted coordinate ring $\mathbb{C}(E)[t,\phi]$, given by $t f = \phi(f)t \forall f \in \mathbb{C}(E)$. Since this is not a commutative algebra, I was looking for the kernel of the function $ \psi: \mathbb{C} \langle X,Y,Z \rangle \rightarrow \mathbb{C}(E)[t,\phi],\\\psi(X) = xt,\psi(Y)=yt,\psi(Z) = 1t,$ where $x = \frac{X}{Z}, y= \frac{Y}{Z}$ and this is where I found my relations for $A$ and $B$ and I was wondering if these algebras pop up somewhere.
Any help would be appreciated.