Let $Spin(p,q)$ be the real spin group to the quadratic space $\mathbb{R}^{p,q}$ defined via Clifford algebras as $Pin(\mathbb{R}^{p,q}) \cap Cl^0(\mathbb{R}^{p,q})$, meaning as the subset of the Clifford algebra which is generated by an even number of elements $v \in \mathbb{R}^{p,q}$ with $Q(v) = \pm 1$, where $Q$ denotes the quadratic form.
Ultimately, in physics we are interested in the irreducible representations of the identity component of the spin group $Spin_0(p,q)$. Now, I would like to understand whether the restriction of an irreducible spinor representation (Induced by the restriction of a representation of the Clifford algebra) of $Spin(p,q)$ remains irreducible.
I tried to prove that an additive basis of the even subalgebra $Cl^0(\mathbb{R}^{p,q})$ is contained in $Spin_0(p,q)$ by showing that the image of the basis under the adjoint representation lies in the identity component $SO_0(p,q)$ of the special orthogonal group, but that did not seem to work.
Could someone prove that these representations remain irreducible or give me a counterexample? Thank you very much in advance!