Regular decompositions of finite root systems and simple Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In this paper we classify all regular decompositions of $\mathfrak{g}$ and its irreducible root system $\Delta$. A regular decomposition is a decomposition $\mathfrak{g} = \mathfrak{g}_1 \oplus \dots \oplus \mathfrak{g}_m$, where each $\mathfrak{g}_i$ and $\mathfrak{g}_i \oplus \mathfrak{g}_j$ are regular subalgebras. Such a decomposition induces a partition of the corresponding root system, i.e. $\Delta = \Delta_1 \sqcup \dots \sqcup \Delta_m$, such that all $\Delta_i$ and $\Delta_i \sqcup \Delta_j$ are closed. Partitions of $\Delta$ with $m=2$ were known before. In this paper we prove that the case $m \ge 3$ is possible only for systems of type $A_n$ and describe all such partitions in terms of $m$-partitions of $(n+1)$. These results are then extended to a classification of regular decompositions of $\mathfrak{g}$.