Linear transformations that preserve the nilpotent matrices (Q1145743)

Let \(\text{sl}_n\) be the algebra of \(n\times n\) matrices with zero trace and entries in a field with at least \(n\) elements. Let \(N\subset \text{sl}_n\) be the set of nilpotent matrices. The group of nonsingular linear transformations \(L\) on \(\text{sl}_n\) such that \(L(N) =N\) is generated by the inner automorphisms: \(X\to S-^{-1}XS\) and the transformation: \(X\to X^t\) that sends every matrix to its transpose. The techniques in the proof involve tangent spaces and the fundamental theorem of projective geometry.