Axes in non-associative algebras

"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension $d$ of any algebra $A$ generated by a pair $a,b$ of (not necessarily Jordan) axes of respective types $(\lambda,\delta)$ and $(\lambda',\delta')$ must be at most $5$; $d$ cannot be $4.$ If $d\le 3$ we list all the possibilities for $A$ up to isomorphism. We prove a variety of additional results and mention some research questions at the end.