Exercise 12, § 4

Let $G$ be a real or complex connected finite-dimensional Lie group, $\alpha$ a differential form of degree $p$ invariant on the left on G. For $\alpha$ to be invariant on the right as well, it is necessary and sufficient that $d\alpha = 0$. (Observe that the condition of invariance on the right is written

for all $s \in G$, and is therefore equivalent to the condition

whatever $u, u_1, \ldots, u_p$ in $L(G)$ may be). In particular, the differential forms of degree 1 invariant on the right and on the left on $G$ form a vector space of dimension $\dim L(G) - \dim[L(G), L(G)]$.