Orthogonal decomposition of some affine Lie algebras in terms of their heisenberg subalgebras

In the present note we suggest an affinization of a theorem by Kostrikin et. al. about the decomposition of some complex simple Lie algebras $\mathcal G$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac–Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r$, $C_{2^m}$, $D_r$, $G_2$, $E_7$, $E_8$ can be decomposed into the algebraic sum of pairwise orthogonal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.