Remark 1

Since the algebra $C(Q)$ is simple, it has only one class of irreducible representations; they are called spinor representations; when one fixes one's attention on one of these representations, say $\tau$, the elements of the space on which $\tau$ is carried out are called spinors. If Q is neutral, the restriction of $\tau$ to $C^+(Q)$ is, like that of $s$, the sum of two inequivalent absolutely irreducible representations; the elements of the subspaces on which these two representations are carried out are called semispinors. In the general case, if $C^+(Q)$ is not simple, the restriction of $\tau$ to $C^+(Q)$ must, since it is faithful, contain subrepresentations belonging to each of the two classes of irreducible representations of $C^+(Q)$, and hence is the sum of two inequivalent absolutely irreducible representations, since this is so after extension of scalars to the algebraic closure $A'$ of $A$. On the other hand, if $C^+(Q)$ is simple, it has only one class of irreducible representations, and the restriction of $\tau$ to $C^+(Q)$ is therefore irreducible, since it decomposes by extension of scalars to $A'$ into two non-equivalent representations.