anti-involution (the Rosati involution) $b\mapsto b^\dagger$ which is positive in the sense that for every nonzero $b\in {\operatorname{End}}_{{\mathbb Q}}(A)$, ${\operatorname{tr}}(bb^\dagger)> 0$. This remains so after tensoring with ${{\mathbb R}}$. We recall a few facts about such algebras.

Given a finite dimensional ${{\mathbb R}}$-algebra ${{\mathscr B}}$ with a positive anti-involution $\dagger$, then by the classification of such algebras, ${{\mathscr B}}$ naturally decomposes into a product of ordinary matrix algebras with coefficients ${{\mathbb R}}$, ${{\mathbb C}}$, or ${{\mathbb K}}$, and $\dagger$ is then in each factor equivalent to taking the conjugate transpose. So the set of $f\in {{\mathscr B}}$ with $ff^\dagger=1$ is a subgroup $L ({{\mathscr B}})$ of the group of units in ${{\mathscr B}}$. It is a finite extension of a product of orthogonal groups, unitary groups and unitary quaternion groups, so in particular, compact. Moreover, it generates ${{\mathscr B}}$ as an ${{\mathbb R}}$-algebra. Given a finite dimensional ${{\mathscr B}}$-module $F$, then we say that an element of ${\operatorname{Sym}}^2F$ is ${{\mathscr B}}$-invariant if for all $b\in {{\mathscr B}}$, it is killed by $b\otimes 1-1\otimes b^\dagger$. Since $L({{\mathscr B}})$ generates ${{\mathscr B}}$ as an algebra, this is equivalent to this element being $L({{\mathscr B}})$-invariant in the usual sense. There is similar notion 