The twisted Clifford algebra

Question

Let $(E,g)$ be a vector space with metric $g$ and let $\alpha \in Gl_n(E)$ be an automorphism of $E$, then the twisted Cliffod algebra is:

$$e\otimes \alpha (f)+f \otimes \alpha (e)=2g(e,f)$$

Can we define the corresponding spinors for the twisted Clifford algebra?