  We develop a Borel-de Siebenthal theory for affine reflection systems by
classifying their maximal closed subroot systems. Affine reflection systems
(introduced by Loos and Neher) provide a unifying framework for root systems of
finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras,
and extended affine Lie algebras. In the special case of nullity $k$ toroidal
Lie algebras, we obtain a one-to-one correspondence between maximal closed
subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a
prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and
$H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This
generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie
algebras.
