Strong commutativity preserving maps of strictly triangular matrix Lie algebras

Let N(F) be the Lie algebra consisting of all strictly upper triangular (n + 1) x (n + 1) matrices over a field F. An invertible linear map phi on N(F) is called to be strong commutativity preserving (simply denoted by SCP) if [phi(x), phi(y)] = [x, y] for any x, y is an element of N(F). We show that for n >= 4, an invertible linear map phi preserves strong commutativity if and only if there exist b, b(1), b(2) is an element of F and a linear function f :N(F) -> F with f(E-1,E-n+1) not equal -1 such that phi= +/- sigma(b)mu((n,n+1))(b2) mu((12))(b1)phi f, where sigma(b), is an inner SCP, mu((n,n+1))(b2), mu((12))(b1) are extremal SCPs, phi f is a central SCP.