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For each two-dimensional vector space V of commuting n×n matrices over a field F with at least 3 elements, we denote by V˜ the vector space of all (n+1)×(n+1) matrices of the form [A¿00] with A¿V. We prove the wildness of the problem of classifying Lie algebras V˜ with the bracket operation [u,v]:=uv-vu. We also prove the wildness of the problem of classifying two-dimensional vector spaces consisting of commuting linear operators on a vector space over a field.Postprint (author's final draft

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Let F be a field F of characteristic zero. Let W n (F) be the Lie algebra of all F-derivations with the Lie bracket [D 1 , D 2 ] := D 1 D 2 − D 2 D 1 on the polynomial ring F[x 1 , . . . , x n ]. The problem of classifying finite dimensional subalgebras of W n (F) was solved if n ≤ 2 and F = C or F = R. We prove that this problem is wild if n ≥ 4, which means that it contains the classical unsolved problem of classifying matrix pairs up to similarity. The structure of finite dimensional subalgebras of W n (F) is interesting since each derivation in case F = R can be considered as a vector field with polynomial coefficients on the manifold R n .Dedicated to the 70-th birthday of Professor V.V. Sergeichuk

Abstract. Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerA of all K-derivations of A is an A-module in a natural way and if R is the quotient field of A then RDerA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerA of rank k over R (i.e. such that dim R RL = k), then the derived length of L is at most k and L is finite dimensional over its field of constants. In case of solvable Lie algebras over a field of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector fields with polynomial, rational, or formal coefficients.