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Important subalgebras of a Lie algebra of an algebraic group are its toral subalgebras, or equivalently (over fields of characteristic 0) its Cartan subalgebras. Of great importance among these are ones that are split: their action on the Lie algebra splits completely over the field of definition. While algorithms to compute split maximal toral subalgebras exist and have been implemented [Ryb07, CM09], these algorithms fail when the Lie algebra is defined over a field of characteristic 2 or 3. We present heuristic algorithms that, given a reductive Lie algebra L over a finite field of characteristic 2 or 3, find a split maximal toral subalgebra of L. Together with earlier work [CR09] these algorithms are very useful for the recognition of reductive Lie algebras over such fields., Comment: 18 pages, 4 figures

We give an overview of some properties of Lie algebras generated by at most 5 extremal elements. In particular, for any finite graph {\Gamma} and any field K of characteristic not 2, we consider an algebraic variety X over K whose K-points parametrize Lie algebras generated by extremal elements. Here the generators correspond to the vertices of the graph, and we prescribe commutation relations corresponding to the nonedges of {\Gamma}. We show that, for all connected undirected finite graphs on at most 5 vertices, X is a finite-dimensional affine space. Furthermore, we show that for maximal-dimensional Lie algebras generated by 5 extremal elements, X is a point. The latter result implies that the bilinear map describing extremality must be identically zero, so that all extremal elements are sandwich elements and the only Lie algebra of this dimension that occurs is nilpotent. These results were obtained by extensive computations with the Magma computational algebra system. The algorithms developed can be applied to arbitrary {\Gamma} (i.e., without restriction on the number of vertices), and may be of independent interest., Comment: 19 pages

Let L be the Lie algebra of a simple algebraic group defined over a field F and let H be a split Cartan subalgebra of L. Then L has a Chevalley basis with respect to H. If the characteristic of F is not 2 or 3, it is known how to find it. In this paper, we treat the remaining two characteristics. To this end, we present a few new methods, implemented in Magma, which vary from the computation of centralisers of one root space in another to the computation of a specific part of the Lie algebra of derivations of $L$., Comment: 22 pages

Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type., Comment: 11 pages

The long-root elements in Lie algebras of Chevalley type have been well studied and can be characterized as extremal elements, that is, elements $x$ such that the image of $(\ad x)^2$ lies in the subspace spanned by $x$. In this paper, assuming an algebraically closed base field of characteristic not 2, we find presentations of the Lie algebras of classical Chevalley type by means of minimal sets of extremal generators. The relations are described by simple graphs on the sets. For example, for $C_n$ the graph is a path of length $2n$, and for $A_n$ the graph is the triangle connected to a path of length $n-3$., Comment: 26 pages, 6 figures