  $\pi$-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, following
the work of Naito we provide examples illustrating how Borcherds Kac-Moody
algebras can also be embedded inside a rank $2$ Kac-Moody algebra.
