If $f : X \rightarrow Y$ is a homomorphism of Lie groups, and $f_*$ is the derivative of $f$ at the identity, what is $f_{*,x}$, for some element $x \in X$? This notation isn't in the book I'm using (Brocker and tom Dieck use $Lf$ instead of $f_*$), but this notation is in a problem set and I'm not sure what it means.
It looks like $f_{*, x}$ is a map from the Lie algebra of $X$ to the Lie algebra of $Y$, so my best guess is that $f_{*, x} = (f \circ l_x)_*$, where $l_x$ is left-translation by $x$.