Invariant volume form on a Lie group (${\omega }_g=(L_{g^{-1}}{)}^{\ast }{\omega }_e$)

Choose a nonzero element of ${\Lambda }^n{T}_e^{\ast }G$ for an $n$-dimensional Lie group and transport it by inverse left translations. This gives a smooth nowhere-zero left-invariant differential form of top degree, hence an orientation. The identity $L_{(hg{)}^{-1}}{L}_h=L_{g^{-1}}$ proves invariance. The space of invariant top forms is one-dimensional, since evaluation at the identity identifies it with ${\Lambda }^n{T}_e^{\ast }G$.