Lemma 1

Let G be a Lie group, K a compact subgroup of G, and F an invariant bilinear form on L(G). Let $x, y\in L(G)$. There exists an element $k$ of K such that $F(u$,[(Ad $k$)$(x), y]) = 0$ for all $u\in L(K)$.

The function $v \rightarrow$ F((Ad $v$)$(x), y$) from K to $\mathbf{R}$ is continuous, and hence has a minimum at some point $k\in K$. Let $u\in L(K)$ and put

$h(t) =$ F((Ad exp($tu$)$.k$)$(x), y$)$,t\in \mathbf{R}$.

We have $h(t)\geq h(0)$ for all $t$; moreover, by Chap. III, §3, no. 12, Prop. 44,

$\frac{dh}{dt}(0) = F([u$,(Ad $k$)$(x)], y) = F(u$,[(Ad $k$)$(x), y])$,

hence the lemma (Functions of a Real Variable, Chap. I, §1, no. 7, Prop. 7).