I am seeing examples of riemannian metrics and in our course we define a metric on a Lie group as follows ($^*$ is pullback)
$$g_a={L_{a^{-1}}}^*g_e$$
Where
$$L_a:G\rightarrow G:b\mapsto ab $$ and $g_e$ is a scalar product over $T_eM$.
Just under the line defining $g_a$ teacher rights $$ g_a(X,Y) = g_e({L_{a^{-1}}}_{*a}(X),{L_{a^{-1}}}_{*a}(Y)) $$
I suppose it should have actually been $$g_a=({L_{a^{-1}}}^*g)_a$$ This makes more sense but I want to be sure my definitions are right and I haven't found anything on the internet.
Remember the definition of the pullback of a form: for $\omega : \mathcal{M} \to T_{\, \, \, 2}^{0}(\mathcal{M})$ and an aplication $\varphi : \mathcal{N} \to \mathcal{M}$, $\varphi^{*} \omega: \mathcal{N} \to T_{\, \, \, 2}^{0}(\mathcal{N})$ is defined by $$(\varphi^{*}\omega)_{p}(v,w)=\omega_{\varphi(p)}(\varphi_{*p}(v),\varphi_{*p}(w))$$ for $p \in \mathcal{N}$ and $v,w \in T_{p}(\mathcal{N})$.