I periodically feel like I should learn more about nonassociative algebra. (I’ve studied Lie algebras, and technically Lie algebras are non-associative, but they’re pretty atypical of nonassociative algebras.) There’s a mysterious circle of “exceptional” examples that are all related — the octonions, the five exceptional Lie algebras, the exceptional Jordan algebra — that I would like to understand better. John Baez has an article about the direct connection that I post about before, but what I don’t understand about the general theory is how relaxing assocativity gives you so few new examples.
Project Gutenberg has a book by Schafer on the general theory of nonassociative algebras. Kevin McCrimmon has an unpublished draft of a book on the structure theory of alternative algebras.
I previously linked to an article classifying the simple Moufang loops. The only examples that are not groups are again related to the octonions.