Right-invariant vector field ($r_{\xi }(g)=(d{R}_g{)}_e\xi$)

The right-invariant vector field associated with a Lie algebra element $\xi$ is obtained by right translation of its value at the identity. Its flow is $g\mapsto \exp (t\xi )g$, hence consists of left translations on a Lie group. Right invariance follows because left and right multiplication commute.

For a Matrix Lie group, the right-invariant vector field is $r_{\xi }(g)=\xi g$. Differentiating these ambient matrix-valued functions in the definition of the Lie bracket of vector fields gives $[r_{\xi },r_{\eta }](g)=(\eta \xi -\xi \eta )g$. Thus $\xi \mapsto r_{\xi }$ is an antihomomorphism, and $\xi \mapsto -r_{\xi }$ is a homomorphism. In contrast, $\xi \mapsto l_{\xi }$ for left-invariant vector fields is a homomorphism.