This question might be a little too open-ended, so I will not be chagrined if it is closed and will appreciate suggestions to make it more specific in the comments.
A little background: I recently realized how much I take associativity for granted. I was reading about sedenions, which are a nicely normed $*$-algebra with multiplicative inverses. However, the sedenions have zero divisors: $$st = 0 \text{ does not imply $s$ or $t$ must be zero.} $$ I spent a while struggling to reconcile the existence of multiplicative inverses with the fact that there are multiple solutions $u$ to $su = 0$ (in this case, $u=0$ and $u=t$ both work). The resolution was just that I forgot sedenions were not associative and not even alternative. Previously having (briefly) studied octonions, which are alternative, I could still multiply by inverses safely: $$a^{-1}(ab) = (a^{-1}a)b = b$$ but that is now lost for the sedenions. My question is: What do we gain from the existence of inverses if we do not have associativity or its weaker form of alternativity? To attempt to make the question more specific, is there a structure in the sedenions or other nicely normed $*$-algebras that follows from the existence of inverses even without associativity nor alternativity?