An example of nonsymmetric dipolarizations in a Lie algebra (Q1321756)

Let \({\mathfrak g}\) denote the Lie algebra of real upper triangular matrices. The authors show that there exist subalgebras \({\mathfrak g}^ +\) and \({\mathfrak g}^ -\) of \({\mathfrak g}\) and a linear form \(f\) on \({\mathfrak g}\) such that \({\mathfrak g} = {\mathfrak g}^ + + {\mathfrak g}^ -\), \(f([X, {\mathfrak g}]) = 0\) if and only if \(X \in {\mathfrak g}^ + \cap {\mathfrak g}^ -\), and \(f([{\mathfrak g}^ +, {\mathfrak g}^ +]) = f([{\mathfrak g}^ -, {\mathfrak g}^ - ]) = 0\), where, for \(n \geq 4\), \({\mathfrak g}^ +\) and \({\mathfrak g}^ -\) are not isomorphic.