The Levi-Civita-Lie connection

Question

Let $(E,[,],,\varphi )$ be a Lie fiber bundle (see my last message). A Lie connection is such that:

$$\nabla_s (f.s')= \varphi (s)(f).s' + f.\nabla_s (s')$$

with $s,s'$ sections and $f$ a smooth function.

I define the Levi-Civita-Lie connection over $E$ with metric $g$:

$$\varphi (s'').g(s,s')= g(\nabla_{s''} s,s')+g(s,\nabla_{s''} s')$$

$$\nabla_s s' -\nabla_{s'} s = [s,s']$$

Can we define the Ricci curvature and the Einstein equations?