Take a vector space $V={A,B,C,...}$ (of matrices), and the commutator $[A,B]=AB-BA$, then a Lie algebra of $V$ is characterized by $[V,V]$ staying in $V$. (Loosely speaking.)
What happens when you replace the commutator with some other thing (based on matrix products), again presupposing that the result stays in $V$? With the anticommutator you get Jordan algebras, so at least there are interesting alternatives. Take the most general form with "product dimension" two:
$<A,B>=f_1*AA+f_2*AB+f_3*BA+f_4*BB$
Has the resulting algebra already been researched (or is it pointless for whatever algebraic reason)? In both cases, a reference would come in handy.
If you go to "dimension 3", the most obvious form of a "commutritor" would be
$<A,B,C>=ABC+BCA+CAB-ACB-BAC-CBA$
Again, I'd like to know either a reference or a reason why this is no useful idea.
(Surely someone had that idea before me!)
