  In this paper, we explain how to compute the Lie algebra of the differential
Galois group of a reducible linear differential system. We achieve this by
showing how to transform a block-triangular linear differential system into a
Kolchin-Kovacic reduced form. We combine this with other reduction results to
propose a general algorithm for computing a reduced form of a general linear
differential system. In particular, this provides directly the Lie algebra of
the differential Galois group without an a priori computation of this Galois
group.
