Proposition 37

Let A be a complete normable unital associative algebra and $x \mapsto x^t$ a continuous linear mapping of A into A such that $(x^t)^t = x$ and $(xy)^t = y^t x^t$ for all $x, y$ in A. Suppose that K is of characteristic $\neq 2$. Let G be the subgroup of $A^*$ consisting of the $x \in A$ such that $xx^t = x^t x = 1$. Then G is a Lie subgroup of $A^*$ and $\mathbf{L}(G)$ is the set of $y \in A$ such that $y^t = -y$.

Let S (resp. S') be the set of $y \in A$ such that $y = y^t$ (resp. $y = -y^t$). Then S, S' are closed vector subspaces of A. The formula

proves that A is the topological direct sum of S and S'. Let f be the mapping of A into S defined by $f(x) = xx^t$. This mapping is analytic. For all $y \in A$, $f(1 + y) = 1 + y + y^t + yy^t$; choose a norm on A compatible with its algebra structure; then

Thus, $T_1(f)(y) = y + y^t$, so that f is a submersion at 1. Therefore, there exists an open neighbourhood U of 1 in A such that $U \cap G$ is a submanifold of U. Hence (§ 1, no. 3, Proposition 6) G is a Lie subgroup of A*. Moreover, $L(G) = T_e(G) = \operatorname{Ker} T_1(f)$.