We consider a Lie group G with a bi-invariant semi-Riemannian \({\mathrm{metric}}\ g\) of index \(\nu \) . In the Riemannian case, when \(\nu =0\) in this initial data, K. Sekigawa defined a nearly Kähler structure on the product Lie group \(G \times G\) . In a previous work, the authors investigated some properties of such structure. In this manuscript, we prove that a similar construction works in the more general case for any value of \(\nu \) . So, we get that this product Lie group has a semi-Riemannian nearly Kähler (NK) structure \((G \times G, \langle , \rangle , J )\) whose corresponding NK metric has index \(2 \nu \) . We remark that the corresponding NK metric is not the semi-Riemannian product metric one and we prove a relation between these two metrics. We give a relation between the Levi-Civita connection of the NK metric and the product metric. We calculate the Riemann tensor, the Ricci curvature of such NK structure. In particular we prove that if \((G,g)\) is an Einstein manifold, then \(G \times G\) is also Einstein. It is well known that there are Lie groups that do not admit a bi-invariant Riemannian metric but they admit a bi-invariant semi-Riemannian or a Lorentz one where \(\nu =1\) . For example, we recall some details to see that the Killing form of either the special linear group or semi-orthogonal group induces a bi-invariant semi-Riemannian metric.