The last form follows from the first when we remember that

_g__{μν} _g_^{μ′ν} = δ^{μ′}_{μ} , and therefore _g__{μν}_g_^{μν} =

4,

consequently _g__{μν}_dg_^{μν} + _g_^{μν} _dg__{μν} = 0.