Exercise 7, § 6

Let $(A,\mathscr{B})$ be a natural algebra.

a) There exists at most one involution $\iota$ of $\mathscr{B}$ such that the unique $\mathbf{Z}$-modules homomorphism of A into A such that $b\mapsto \iota (b)$ is an anti-automorphism of A satisfying the condition $\tau_A(bb') = 1$ if $b'=\iota (b)$ and $\tau_A(bb') = 0$ otherwise.

When such an involution $\iota$ exists, $(A,\mathscr{B})$ is said to be a basal natural algebra.

Suppose henceforth that $(A,\mathscr{B})$ is a basal natural algebra.

b) One then has $\iota (b) =b$ for all $b\in \mathscr{B}_0$, as well as the formula

c) The structure constants $\gamma_{bb'}^{b''}$ of A satisfy $\gamma_{bb'}^{\iota(b'')}=\tau_A(bb'b'')$ for $(b, b', b'')$ in $\mathscr{B}$.