We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.

We give an algorithm for constructing a basis and a multiplication table of a finite-dimensional finitely-presented Liering. We apply this to construct the biggest t generator Lie rings that satisfy the n-Engel condition, for (t,n) = (t,2), (2,3), (3,3), (2,4).

1s-e derive i-1 sufficient con&ion for a gtnrra.tingset Of an ideal in the free (uon-associative.Iloll-corrlrllntat,ivc) algebra to bc a Grijlln~r basis.I;eing t,his wc fOrnlulat,e an algorithn~ for coniputiug a hISiS Of il finitely present,ed Lie algebra.=It the end of the paper we discuss tlic practical iriil)lClllc:~lt;ttioll Of the i~lgOritllIn.

By Ado's theorem every finite dimensional Lie algebra over a field of characteristic zero has a faithful finite dimensional representation. We consider the algorithmic problem of constructing such a representation for Lie algebras given by a multiplication table. An effective version of Ado's theorem is proved. 1 Introduction When dealing with the problem of representing finite dimensional Lie algebras on computer, two presentations leap into mind: a presentation by matrices and a presentation by an array of structure constants. In the first presentation the Lie algebra is given by a finite set of matrices fA 1 ; : : : ; An g that form a basis of the Lie algebra. If A and B are two elements of the space spanned by the A i , then their Lie product is defined as [A; B] = A \\Delta B \\Gamma B \\Delta A (where the \\Delta denotes the ordinary matrix multiplication) . The second approach is more abstract. The Lie algebra is a (abstract) vector space over a field F with basis fx 1 ; : : : ; x...