Well, I had thought I was OK with that, but it seems not.

Specifically, on a friend's recommendation, I went to our library and looked in Fulton & Harris:

*Representation Theory*.

There I find that, for the generic Lie group

, the map

is a nice mapping

that fixes the the identity.

This is glaringly obvious, right?

But they, F&H, that is, go on to say this defines a mapping

. Now this step I don't quite see.

Clearly

is the group of all automorphisms

, and, by their (F&H's) construction, then

is a Lie group homomorphism.

What I DON'T get is the the leap from

to

.

Any help out there?