= Neutral invariant metrics on the free two-step Lie algebra on three generators
{title2=$\operatorname{signature}(g)=(3,3)$}
Let $[e_1,e_2]=e_6$, $[e_1,e_3]=e_4$ and $[e_2,e_3]=e_5$, with $e_4,e_5,e_6$ central. The symmetric tensor $S=e_1\otimes e_5+e_5\otimes e_1+e_3\otimes e_6+e_6\otimes e_3-e_2\otimes e_4-e_4\otimes e_2$ is invariant under the <Adjoint representation of a Lie algebra>: the derivation action of each of $e_1,e_2,e_3$ cancels in pairs. For $t\ne0$, $Q=tS+\sum_{j=4}^6a_j e_j\otimes e_j$ has determinant $-t^6$ and inverse metric $g=t^{-1}S(\lambda)-t^{-2}(a_5(\lambda^1)^2+a_4(\lambda^2)^2+a_6(\lambda^3)^2)$ in the dual coframe. Its three hyperbolic planes give the displayed <metric signature>. On a connected <Lie group> this supplies a four-parameter family of <bi-invariant pseudo-Riemannian metrics>. Disconnected components must preserve the tensor separately; the automorphism reversing $e_1,e_4,e_6$ while fixing the other generators reverses $S$ and prevents this extension.
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