ABSTRACT
A system of roots is decomposable if it can be split into two mutually orthogonal subsystems. A system is indecomposable if it is not decomposable. It is easy to see that for decomposable simple-root systems, the simple roots in the two orthogonal subsystems commute, and the entire system of roots splits into two commuting subsets. Because the subalgebras commute, the groups they generate also commute. The general subject of the subalgebras of an algebra is quite complicated. But the principle is a simple one. For each algebra, there is a simplest representation, out of which all the other representations can be built. Each possible transformation of the simplest representation is associated with a different subalgebra.
