$\begin{aligned}
&(I+E)(I - E) = \\
&= I - E + E - E^2 \\
&= I - E^2 = \\
&= I - 0 = I
\end{aligned}$

This lets us expand out $Y$ as:
$\begin{aligned}
&Y = GXG^{-1} \\
&Y = (I + E)X(I + E)^{-1} \\
&Y = (I + E)X(I - E) \\
&Y = IXI -IXE + EXI - EXE \\
&Y = X - XE + EX - EXE
\end{aligned}$

Now we assert that because $E$ is small, $EXE$ is of order $E^2$ and will therefore
vanish. This leaves us with:
$GXG^{-1} = Y = X + [E, X]$

and so the lie bracket is the Lie algebra's way of recording the effect of the
group's conjugacy structure.