Example

If $E$ admits a basis consisting of a single element $e$, $T(E)$ is isomorphic to the polynomial algebra $\mathbf{A}[X]$, and $C(Q)$ is a quadratic extension of $\mathbf{A}$, having as basis $(1,\ u)$, where $u$ is the element $u = \rho(e)$ and satisfies $u^2 = Q(e)$.

Let $T^h$ denote the $h$-th tensor power $\bigotimes^h E$ in $T(E)$, and let $T^+$ (resp. $T^-$) be the sum of the $T^h$ for $h$ even (resp. odd).

Comme $T(E)$ is the direct sum of $T^+$ and $T^-$ and $I(Q)$ is generated by elements of $T^+$, $I(Q)$ is the direct sum of $T^+ \cap I(Q)$ and $T^- \cap I(Q)$, and $C(Q)$ is the direct sum of the two submodules $C^+(Q) = \sigma(T^+)$ and $C^-(Q) = \sigma(T^-)$ (also denoted by $C^+$ and $C^-$). The elements of $C^+$ will be called even (resp. odd). We have the relations

(3) $C^+C^+ \subset C^+, \quad C^+C^- \subset C^-, \quad C^-C^+ \subset C^-, \quad C^-C^- \subset C^+$.

In particular $C^+$ is a subalgebra of $C(Q)$.