I too am super pumped about Lie algebras, and have come up against the issue of how to explain the topic to my less mathematically inclined friends. You have t

I too am super pumped about Lie algebras, and have come up against the issue of how to explain the topic to my less mathematically inclined friends. You have to start with groups. Group structure is something that's both beautiful and something that can be truly appreciated in a 'cocktail party' setting. And you can't convey anything of substance and true about Lie algebras to someone who doesn't know what a group is. Then, once your listener is happy and feeling smarter about his new group theory knowledge, you can try to motivate the idea of a Lie algebra however best fits with your research. This way, you are able to tell your listener something comprehensible, and also hint at what else is out there. At least, I've found that this explanation keeps people pretty happy.