Invertible linear maps on Borel subalgebras preserving zero Lie products

Let gg be a simple Lie algebra of rank l   over an algebraically closed field of characteristic zero, bb a Borel subalgebra of gg. An invertible linear map φ   on bb is said preserving zero Lie products in both directions if for x,y∈bx,y∈b, [x,y]=0[x,y]=0 if and only if [φ(x),φ(y)]=0[φ(x),φ(y)]=0. In this paper, it is shown that an invertible linear map φ   on bb preserving zero Lie products in both directions if and only if it is a composition of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism, which extends the main result in [8] from a linear solvable Lie algebra to far more general cases.