Let $G$ be a Lie group with identity element $e$, [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, Lie bracket $[\cdot,\cdot]:\mathfrak g\times\mathfrak g\to\mathfrak g$, exponential map $\exp:\mathfrak g\to G$, and adjoint representation $\operatorname{Ad}:G\to GL(\mathfrak g)$. Let $k\ge 1$, and let $p:\mathfrak g\to \mathbb R$ be a [homogeneous polynomial](/page/Homogeneous%20Polynomial) of degree $k$ satisfying $p(\operatorname{Ad}_g X)=p(X)$ for every $g\in G$ and every $X\in\mathfrak g$. Let $P:\mathfrak g^k\to \mathbb R$ denote the symmetric $k$-linear form associated to $p$ by polarization, so that $p(X)=P(X,\dots,X)$ for every $X\in\mathfrak g$. Then, for every $Y,X_1,\dots,X_k\in\mathfrak g$,

\begin{align*} \sum_{j=1}^k P(X_1,\dots,X_{j-1},[Y,X_j],X_{j+1},\dots,X_k)=0. \end{align*}