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The subspace spanned by $e_4,e_5,e_6$ is the center of a Lie algebra. Each of their tensor squares is therefore invariant. On tensor products the infinitesimal Adjoint representation of a Lie algebra acts by a derivation, $D_x(u\otimes v)=[x,u]\otimes v+u\otimes [x,v]$. Write $u\odot v=u\otimes v+v\otimes u$ and let $S=e_1\odot e_5+e_3\odot e_6-e_2\odot e_4$. The only potentially nonzero checks areThe remaining generators are central, so they act trivially. This proves every claimed tensor invariant.

These tensors are contravariant. To obtain an ordinary metric tensor, take an invertible linear combination and invert it. For arbitrary real $a_4,a_5,a_6$ and $t\ne 0$, setIn the ordered basis $(e_1,e_2,e_3;e_4,e_5,e_6)$ its matrix isSince $K^{T}K=I$, $\det Q=-t^6\ne 0$, andLet ${\lambda }^a$ be the dual left-invariant coframe. The resulting neutral invariant metrics on the free two-step Lie algebra on three generators areHere $({\lambda }^a{)}^2={\lambda }^a\otimes {\lambda }^a$. Infinitesimal invariance of $Q$ implies that its inverse is an adjoint-invariant bilinear form. Its left-translated metric is right-invariant on a connected $G$, giving the desired four-parameter family of bi-invariant metrics. A change of variables removes the central $A$ block from the quadratic form, leaving three off-diagonal two-dimensional blocks. Each has one positive and one negative eigenvalue. Thus the metric signature is $(3,3)$; these are bi-invariant pseudo-Riemannian metrics, not positive-definite ones.

Connectedness, conventionally understood here, matters for the passage from infinitesimal to full group invariance. If $G$ is disconnected, its other components must preserve $Q$ separately. For example the Lie algebra automorphism reversing $e_1,e_4,e_6$ and fixing $e_2,e_3,e_5$ preserves all brackets but sends $S$ to $-S$. In the semidirect product of the simply connected group with this order-two automorphism, none of these nondegenerate $t\ne 0$ tensors is invariant under that component. Thus the Lie-algebra data alone give the stated family on the identity component, rather than an unconditional extension to every disconnected realization.