Left-invariant nearly pseudo-Kähler structures and the tangent Lie group

Let $G$ be a Lie group, and let $(g,J)$ be a left-invariant almost pseudo-Hermitian structure on $G$. It is shown that if $(g,J)$ is also nearly pseudo-Kähler, then the tangent bundle $TG$ (with its natural Lie group structure induced from $G$) admits a left-invariant nearly pseudo-Kähler structure.