I would like an explanation about how commutator identities work.
I am doing Shankar's Principle of Quantum Mechanics book, and the first chapter is all about Linear Algebra. He gives to my hands the following commutator identities:
$$ \Omega \Lambda - \Lambda \Omega \equiv [\Omega, \Lambda]$$
$$ (i) \; [\Omega, \Lambda \theta] = \Lambda[\Omega, \theta] + [\Omega, \Lambda]\theta $$
$$ (ii) \; [\Lambda \Omega, \theta] = \Lambda[\Omega, \theta] + [\Lambda, \theta]\Omega $$
Let me say what I have understood from it.
$(i)$ The commutator between an operator and a product of two operators is equal to the commutator between the left side operator of the original commutator and the second operator of the right side, multiplied by the first operator of the right side of the original commutator, plus the commutator between the operator of the left side of the original commutator and the first operator of the right side of the original commutator, multiplied by the second operator of the right side of the original commutator.
$(ii)$ You should have got an idea what it should looks like from what I've written in item $(i)$.
What I have understood is correct? If not, what would be the correct interpretation of this identity? If yes, is there any other way to explain it simpler?
Thank you in advance!