A description of ad-nilpotent elements in lie algebras arising from semiprime rings with involution

We study ad-nilpotent elements in Lie algebras arising from semiprime rings R without 2-torsion. In order to keep under control the torsion of R we introduce a more restrictive notion of ad-nilpotence, pure ad-nilpotence, which is a technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows the torsion inside the ring R to be more accurate. If R is a semiprime ring and a ∈ R is a pure ad-nilpotent element of R of index n with R free of t and ( n t ) -torsion for t := [ 2 ], then n is odd and there exists λ ∈ C(R) such that a − λ is nilpotent of index t. If R is a semiprime ring with involution ∗ and a is a pure ad-nilpotent element of Skew(R, ∗) free of t and ( n t ) -torsion for t := [ 2 ], then either a is an ad-nilpotent element of R of the same index n (this may occur if n ≡ 1, 3 (mod 4)) or a is a nilpotent element of index t+ 1 and R satis es a nontrivial GPI (this may occur if n ≡ 0, 3 (mod 4)). The case n ≡ 2 (mod 4) is not possible.