well understood, but the corresponding decompositions may nevertheless provide valuable new viewpoints.

We will use the following theorem, which allows us to find subalgebras by locating a set of simple roots for the subalgebra within the root system of the larger algebra. One cannot just choose an arbitrary subset of positive roots and declare them to be the simple roots of a subalgebra. For example, if root vectors $E_i$ correspond to simple roots $\beta_i$ and root vectors $F_j$ correspond to their negatives $-\beta_j$, then one of the required Serre relations, $[E_i,F_j] = \delta_{ij} H_i$ could be violated for $i\neq j$ if $\beta_i - \beta_j$ were a root of the larger algebra.

\[subrootsys\] Let $\Phi$ be the set of roots of a Kac-Moody Lie algebra, ${{\mathfrak{g}}}$, with Cartan subalgebra, ${{\mathfrak{h}}}$, and let $\Phi_{real}^+$ be the positive real roots of ${{\mathfrak{g}}}$. Let $\beta_1,\cdots,\beta_n\in\Phi_{real}^+$ be chosen such that for all $1\leq i \neq j \leq n$, we have $\beta_i - \beta_j\notin \Phi$. For $1\leq i\leq n$ let $0\neq E_i\in{{\mathfrak{g}}}_{\beta_i}$ and $0\neq F_i\in{{\mathfrak{g}}}_{-\beta_i}$ be root vectors in the one-dimensional root spaces corresponding to the positive real roots $\beta_i$, and the negative real roots $-\beta_i$, respectively, and let $H_i = [E_i,F_i]\in{{\mathfrak{h}}}$. Then the Lie subalgebra 