The transformations of spinors

We begin showing that for even dimensional vector spaces V all automorphisms of their Clifford algebras are inner. So all orthogonal transformations of V are restrictions to V of inner automorphisms of the algebra. Thus under orthogonal transformations P and T - space and time reversal - all algebra elements, including vectors v and spinors \varphi, transform as v \to x v x^{-1} and \varphi \to x \varphi x^{-1} for some algebra element x. We show that while under combined PT spinor \varphi \to x \varphi x^{-1} remain in its spinor space, under P or T separately \varphi goes to a different spinor space and may have opposite chirality. We conclude with a preliminary characterization of inner automorphisms with respect to their property to change, or not, spinor spaces.