The calculation of connection from the first structure equations (Q757847)

The author obtains convenient explicit formulae that produce the connection forms \(\Gamma_{AB}=\Gamma_{BA}\), \(A,B=0,1\), of the torsion free metrical connection of a four-dimensional pseudo-Riemannian (Lorentzian) space from S-forms that satisfy Cartan's first structure equations \[ dS_{AB}+2\Gamma^ C_{(A}\bigwedge S_{B)C}=0 \] and the quadratic identity \[ S_{AB}\bigwedge S^{CD}=\frac{1}{3}\delta^ c_{(A}\delta^ D_{B)}S_{KL}\bigwedge S^{KL}. \]