§ Lie bracket as linearization of conjugation
Let us have with all of these as matrices. Let's say that
is very close to the identity: with ( for epsilon).
Note that now, , which by abuse of notation
can be written as , which by taylor expansion is equal to
. Since is nilpotent, we truncate at
leaving us with as the inverse of . We can check that this is correct, by computing:
This lets us expand out as:
Now we assert that because is small, is of order and will therefore
vanish. This leaves us with:
and so the lie bracket is the Lie algebra's way of recording the effect of the
group's conjugacy structure.