SCHWEITZER 2013 Problem 5

A subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak g$ is said to have the $\gamma$ property with respect to a scalar product ${\langle \cdot,\cdot \rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$ . Prove that the maximum dimension of ${\gamma}$ -property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product.

Proposed by Péter Nagy Tibor