I want to classify all metric two-step Lie algebras of dimension 7. Clearly, the dimension of the center of such Lie algebras is $\leq 5$. So, I have to consider 5 cases separately, where the dimension of center is $1,2,\cdots \text{or}\, 5$. Let $\mathfrak{g}$ is a metric seven dimensional Lie algebra and $\mathfrak{h}$ is its one dimensional center. We consider $\mathfrak{a}_3$ is a $3-$dimensional vector subspace of $\mathfrak{g}$ such that $[\mathfrak{a}_3 ,\mathfrak{a}_3]=\mathfrak{h}$. There is a unique vector subspace $\mathfrak{b}_3$ of $\mathfrak{a}$ which is complementary subspace of $\mathfrak{a}_3$ and commutes with $\mathfrak{a}_3$. Then one can choose $\{e_1,\cdots ,e_7\}$ an orthonormal basis of $\mathfrak{g}$ such that $\mathfrak{h}=\text{span}\langle e_7\rangle$ and $\{e_1,e_2,e_3\}$ is an orthonormal basis of $\mathfrak{a}_3$. We have $$[e_i,e_j]=\lambda_{ij} e_7,\qquad i,j\in \{1,\cdots,6\}$$ where $0\neq \lambda_{ij}=-\lambda_{ji}$. I want to find a new orthonormal basis to show that $\mathfrak{g}$ is metric Heisenberg algebra.
Any suggestion is highly appreciated?