The one thing that gave me deep insights into algebra was discovering Peano's axioms.[0] I had been taught the associative, transitive, and distributive laws in 6th grade at school, but discovering the axioms really opened my eyes in high school; especially under what circumstances they could and couldn't be applied. The last penny to drop was to realise that where you have numbers (e.g. x) you could also substitute formulas, provided you adhered to the rules of when the axioms could be applied. Obviously for similar reasons this also applies to matrices (e.g. when can matrices be inverted (additive and multiplicative), and hence why matrix addition is commutative, but multiplication is not).

Don't get me started on subtractive anti-commutativity vs divisive anti-commutativity.

Another thing that bothers me is when helping the my own kids with maths I came to realise that their teachers don't know the Peano axioms, so how the heck can my kids even stand a chance at properly learning algebra.

[0] https://en.wikipedia.org/wiki/Peano_axioms