= Infinitesimal left-action sign convention
{title2=$[X^\#,Y^\#]=-[X,Y]^\#,\quad v_X=-X^\#$}
For a smooth left <Lie group action>, define the <fundamental vector field> by $X^\#(p)=\left.\frac d{ds}\right|_0\exp(sX)\cdot p$. With the <Lie bracket of vector fields> $[U,V]f=U(Vf)-V(Uf)$, this map is an anti-homomorphism. The map $X\mapsto v_X=-X^\#$ is a <Lie algebra representation>. A right action with the positive exponential has the opposite homomorphism convention, so specifying action side and bracket convention prevents a sign ambiguity.
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