It's interesting that so many of these tricks are specifically for testing invariance properties. I wonder if it's because those sort of questions pop up a lot, or maybe those are the type of problems that TT enjoys, or that those problems are particularly amenable to coming up with tricks to solve.

> * Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators

This is the insight that led me to really start to understand linear algebra while working through Axler's Linear Algebra Done Right.

> * Invariance under tensor products -> suffices to check one-dimensional/irreducible case

I suppose this is case dependent, but I can imagine some groups irreducible representations are way messier to handle than others!