I am searching for a proof of the claim made in this post. It states that Majorana spinors (I refer to both complex pinor and spinor representations which are restricted to the real Spin group and admit a $Spin_0(p,q)$-equivariant real structure as Majorana spinors) exists whenever either the Clifford algebra or its even subalgebra has $\mathbb{R}$ as its maximal commutant.
I am searching for a direct proof without the use of coordinates, the use of the complex conjugate representation (as Trautmann does it) or equivariant forms on the spinor modules (I have trouble understanding Varadarajan).
My problem is mainly to understand why the real spinor representations of the real spin group not all induce real structures ($Cl(V,Q) \otimes \mathbb{C} \cong Cl(V_\mathbb{C},Q_\mathbb{C})$, so any real algebra representation $S$ induces a complex algebra representation $S \otimes \mathbb{C}$ which canonically has a real structure, which should be equivariant with respect to the spin action. By dimension counting, we see that these induced representations are equivalent to one of the complex irreducible spinor representations.).
