The Lie bracket of fundamental vector fields

 P: 3 1. The problem statement, all variables and given/known data The Lie bracket of the fundamental vector fields of two Lie algebra elements is the fundamental vector field of the Lie bracket of the two elements: $[\sigma(X),\sigma(Y)]=\sigma([X,Y])$ 2. Relevant equations Let $\mathcal{G}$ a Lie algebra, the fundamental vector field of an element $X\in\mathcal{G}$ is defined at a point $p\in M$ of a manifold $M$ as: $\sigma_{p}(X)=(p\,e^{tX})'(0)$ 3. The attempt at a solution $[\sigma(X),\sigma(Y)](f) = \sigma(X)[\sigma(Y)f]-X\leftrightarrow Y$ $= \sigma(X)[f(pe^{tY})'(0)]-X\leftrightarrow Y$ $= f(pe^{tX}e^{tY})'(0)-X\leftrightarrow Y$ $\sigma([X,Y])(f) = f(pe^{t[X,Y]})'(0)$