= Right-invariant vector fields realize the opposite Lie algebra
{title2=$[r_\xi,r_\eta]=-r_{[\xi,\eta]}$}
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.
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