Toral rank one Lie algebras

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Let L be a finite-dimensional Lie algebra over an algebraically closed field F of characteristic p > 0. Assume that H is a Cartan subalgebra of L. We say that L has toral rank m relative to H if dim,PA = m, where A is the set of roots with respect to H, P is the prime field, and PA is the P-vector space spanned by A. In this work we study Lie algebras of toral rank one. The motivation to investigate this class of algebras comes from the problem of classifying simple Lie algebras. If A is a finite dimensional simple Lie algebra with Cartan subalgebra H, and if CI is a root of H, then the subalgebra A(” = CieP A, has toral rank one relative to H. In [12, 133 Wilson has shown that if a simple algebra A has characteristic p > 7, then the restricted closure w of H in Der A has the form R= T+ Z, where I is the nil radical of B and T is the maximal torus, which is central in j?. It follows that ad R on each root space A, can be simultaneously upper triangularized. Indeed, I is a Lie algebra of nilpotent transformations on A,; and thus by [8, Theorem 3.31, it annihilates a nonzero vector. The space of such vectors is T-invariant and so T + I has a common eigenvector u on A,. One can proceed inductively on AJFu to argue that each root space A, has a basis relative to which ad R is upper-triangular. With that in mind, we say in what follows that a Cartan subalgebra His triangularizable if there is a basis for each root space relative to which ad H is upper-triangular. We show in Section 1 that if L is a Lie algebra having a toral rank one Cartan subalgebra H, and if H is triangularizable, then L modulo its maximal solvable ideal either is (0) or consists of a simple toral rank one Lie algebra S together with derivations of S in H. In the first case we say that the core of L is (0) and in the second case that the core of L is S. Wilson [ 141 has classified the simple toral rank one Lie algebras. Using