I have (what I find) a handy mnemonic for the law of total variance. It’s also sometimes called the conditional variance formula, or (the reason for this mnenomic) Eve’s law. At least if I’m quoting Wikipedia right.
I remember it as “e-vee plus vee-e”.
By applying the functions pointwise, you can write the law as
\[\mathbb{V}(X) = (\mathbb{EV} + \mathbb{VE} )(X | Z)\]Where the expectation and variance are nested functions that are each applied to \(X|Z)\) and then summed. It’s also fun to say.
It’s apparently also why it’s called Eve’s law in the first place, going off this Quora post.