The spinorial Clifford algebra

Question

Let $(E,g)$ be a metric vector space with Clifford algebra $Cliff(E,g)$ and the spinor space $(\Sigma (E,g), g')$ ; $g'$ is the real part of the hermitian product. Then I define the spinorial Clifford algebra $Cliff(\Sigma,g,g')$ by the following relations:

$$(X.\psi)(Y.\psi ')+(X.\psi')(Y.\psi)+(Y.\psi)(X.\psi')+(Y.\psi')(X.\psi)=4 g(X,Y)g'(\psi,\psi')$$

with $X,Y$ in $E$ and $\psi,\psi'$ in $\Sigma$.

Can we study the representations of the spinorial Clifford algebra?