  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 $\mathfrak{g}$.
