\(\pi \) -Systems are fundamental in the study of Kac–Moody Lie algebras since they arise naturally in the embedding problems. Dynkin introduced them first and showed how they also appear in the classification of semisimple subalgebras of a semisimple Lie algebra. In this article, we explicitly classify the \(\pi \) -systems associated to rank 2 Kac–Moody Lie algebras and prove that in most of the cases they are linearly independent. This classification allows us to determine the root generated subalgebras and which in turn determines all possible Kac–Moody algebras that can be embedded in a rank 2 Kac–Moody algebra as subalgebras generated by real root vectors. Additionally, we also discuss by examples that classification of \(\pi \) -systems is quite challenging even for rank 2 Kac–Moody algebras if we allow imaginary roots in \(\pi \) -systems.
