Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$
Set-Up: The type of Lie algebra of our interests (found out during a physics research) have $e_a$ with $a=1,2,3,4$, four generators. The form of structure constant is given by $$ [e_3,e_i]=\epsilon_{ij} e_j,\;\;\; [e_i,e_j]=\epsilon_{ij} e_4,\;\;\;\;\; $$ (with generators $e_j$ of Lie group) and others Lie brackets (commutators) are zeros. Here $i, j$ are only $1,2$.
The Killing form of this algebra can be explicitly computed as a degenerate Killing form: $$ g_{ab}=\begin{pmatrix} 0 & 0& 0& 0\\ 0 & 0& 0& 0\\ 0 & 0& 2& 0\\ 0 & 0& 0& 0 \end{pmatrix} $$
Question: How to determine/construct non-degenerate invariant bilinear form $\Omega_{ab}$ for this Lie algebra?
Please be specific to focus on this special case (instead of being vague on outlining the general procedure). Many thanks!