Left-invariant differential form

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.