Direct factor of a loop

Definition
A direct factor of an algebra loop $$L$$ is a subloop $$S$$ such that there exists a subloop $$T$$ of $$S$$ satisfying:


 * Every element of $$L$$ can be written uniquely in the form $$s * t, s \in S, t \in T$$.
 * For any $$s_1,s_2 \in S, t_1, t_2 \in T$$, we have $$\! (s_1 * t_1) * (s_2 * t_2) = (s_1 * s_2) * (t_1 * t_2)$$. In particular, by an idea analogous to the Eckmann-Hilton principle, every element of $$S$$ commutes with every element of $$T$$.