Remark

(1) Let $E^+$ be the graded subalgebra of E the direct sum of the $E_{2n}$ ($n \in \mathbf{Z}$); it follows from Definition 7 that if E is anticommutative, $E^+$ is a subalgebra contained in the centre of E (and hence commutative).

(2) Suppose that 2 is not a divisor of 0 in E; then if E is anticommutative E is alternating, since for $x \in E$ homogeneous and of odd degree, $x^2 = -x^2$ by (36), whence $2x^2 = 0$ and $x^2 = 0$ by virtue of the hypothesis.

(3) We shall study in detail in § 7 important examples of alternating algebras.