The Lie algebra version of the Krull-Schmidt Theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. For finite-dimensional Lie algebras, there is a well-known algorithm for finding such components, so the theorem considerably simplifies the problem of classifying the automorphism groups. We illustrate this by classifying the automorphisms of all indecomposable real Lie algebras of dimension five or less. Our results are presented very concisely, in tabular form.

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Computing the Lie point symmetries of systems of linear differential equations can be prohibitively difficult. For homogeneous systems in Kovalevskaya form of order two or higher, this paper proves the existence of a basis of infinitesimal generators (as determined by Lie's algorithm) whose characteristic forms are homogeneous in the dependent variables of degree zero, one or two. Suppose Lie's algorithm yields a characteristic form of degree two; in this case, the system is second order. If it contains only ordinary differential equations (ODEs), its general solution is constructed from those of a given first-order linear homogeneous system, of the same dimension, and a second-order linear scalar equation. Otherwise, coordinates are given in which its dimension (necessarily two or higher now) is lowered by one, leaving an inhomogeneous system of parametrized ODEs. Its homogeneous part is solved as in the previous case.

The systematic method for calculating all point symmetries of partial differential equations (PDEs) with non-trivial Lie point symmetry algebras does not currently apply to linear PDEs. Consider any given locally solvable system of linear homogeneous PDEs, each of order 2 or higher, of general Kovalevskaya form. Suppose that its Lie point symmetry generators' characteristics are linear in the dependent variables, while those that are homogeneous of degree 1 in the dependent variables contain a finite number of arbitrary parameters (this is common for systems that arise from applications). Herein, every point symmetry is shown to be projectable/fibre-preserving: the transformed independent variables depend only on the original independent variables. Moreover, two (conditional) Lie-algebraic characterizations of the generators of scalings and superpositions of solutions are derived. One of them holds if the system cannot be decoupled (with a smooth, locally invertible, linear change of dependent variables). If any such characterization holds, every point symmetry maps the dependent variables to functions that are linear in the original dependent variables. The aforementioned systematic method is adapted to calculate these symmetries. This version of the method also applies to linearizable systems.

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