It is hard to explain the value of higher math, but it's worth it.

Perhaps a better answer to "What are Lie algebras?" is to respond in terms that mean something to your audience. Avoid words like "vector" and "combinatorics".

Instead use metaphors. Like Rubix cubes. Tell them Lie algebra is a way to solve Rubix cubes faster. And also other similar puzzles that are way harder than Rubix cubes. It's true enough for casual conversation and probably more interesting than a vaguer answer.