Found a very nice series on Lie group and Lie algebra. Watched the first couple lectures. The main message I got so far is that Lie algebra combine techniques from analysis (calculus) with algebra (group).
And one nice thing I learned is that $latex \exp(A+B) = \exp(A) \exp(B)$ if $latex AB = BA$. I didn’t realize the commutation condition is necessary earlier.
When $latex AB \neq BA$, the Baker-Campbell-Hausdroff formula said that
$latex \exp(A) \exp(B) = \exp(A+B+\frac{1}{2}[A,B]+\underset{\mbox{only depends on brackets}}{\underbrace{\cdots}})$