Solution
= Solution
The <Lie bracket of vector fields> is bilinear, alternating, satisfies the <Jacobi identity>, and is natural under <diffeomorphisms>. Therefore the bracket of two <left-invariant vector fields> is left-invariant. Define
$$
[\xi,\eta]=[l_\xi,l_\eta]_e.
$$
Then
$$
l_{[\xi,\eta]}=[l_\xi,l_\eta],
$$
and the inherited bilinearity, alternation, and Jacobi identity make $T_eG$ a <Lie algebra>.