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If $f : G\to H$ is a Lie group homomorphism,

  1. what can we say about its differential $d_ef : \mathcal{G}\to\mathcal{H}$?
  2. Is it a Lie algebra morphism or anti-morphism? $$d_ef\left([\xi,\eta]\right)=[d_ef(\xi),d_ef(\eta)]\ \text{or}\ -[d_ef(\xi),d_ef(\eta)]?$$

If $f$ is an isomorphism, then it's clear $$f_*\left([\xi,\eta]\right)=[f_*\xi,f_*\eta]$$ in the general case we can't push-forward a vector field by $f$...

I am thinking about a left action of a Lie group $G$ on a smooth manifold $M$, so a group homomorphism $$\psi : G\to\mathrm{Diff}(M),$$ (but $\mathrm{Diff}(M)$ is not a Lie group!). In this cae, $d_e\psi$ is a Lie algebra anti-morphism from $\mathcal{G}$ to fundamental vector fields.

Thanks for any help!

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