Proposition 4

With the assumptions and notation of Th. 3, assume that K is of characteristic $\neq 2$. Then $-1 \in G$ if and only if the characteristic degrees $k_1, \ldots, k_l$ of R are all even.

Let $f$ be the automorphism of the algebra S that extends the automorphism $-1$ of V. Then $f(z) = (-1)^{\deg z} z$ for all homogeneous $z$ in S. Thus, if $-1 \in G$, every homogeneous element of odd degree of R is zero, and the $k_i$ are all even. Conversely, if the $k_i$ are all even, every element of R is invariant under $f$, and Galois theory shows that $-1 \in G$.