nilpotent Lie algebras above can be constructed as extensions of abelian Lie algebras. This is the way in which they appear in [@FS], where Favre and Santharoubarne obtained the classification of the nilpotent Lie algebras of dimension $\leq  7$ admitting an ad-invariant metric. According to their results, any of the Lie algebras equipped with an ad-invariant metric ${\mathfrak n }_{3,2}$ or ${\mathfrak n }_{2,3}$ as above is equivalent to one of the followings

$$\label{m3223}
({\mathfrak n }_{3,2}, B_{3,2}) : \left(
\begin{matrix}
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
 0 & 0  & 1 & 0 & 0 & 0 \\
 0 & -1 &  0 & 0 & 0 & 0 \\
 1 & 0 & 0 & 0 & 0 & 0
 \end{matrix}
 \right) \;\;
 ({\mathfrak n }_{2,3}, B_{2,3}) : 
 \left(
\begin{matrix}
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & - 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & -1 & 0 & 0 