Left and right translation on a Lie group ($L_g,R_g$)

For a Lie group $G$, left and right translation are the diffeomorphisms $L_g(h)=gh$ and $R_g(h)=hg$. They commute: $L_g{R}_h=R_h{L}_g$.

For $\xi$ in the Lie algebra of $G$, the left-invariant vector field is $l_{\xi }(g)=(d{L}_g{)}_e\xi$. Its global flow is right translation ${\Phi }^t=R_{\exp (t\xi )}$.

A differential form $\alpha$ on a Lie group is left-invariant when $L_g^{\ast }\alpha =\alpha$ for every $g$. It is determined by its value at the identity.

A differential form on a Lie group is bi-invariant when it is invariant under both left and right translations.

On a connected Lie group, a left-invariant differential 1-form is closed exactly when it is bi-invariant. Connectedness is essential: on $O(2)$ every left-invariant 1-form is closed because its Lie algebra is abelian, while reflection conjugation negates every nonzero one.