Definition of "left-invariant connection"

Question

I'm reading a paper which formulates a theorem which holds for a "left-invariant connection" on a Lie group G. I would guess that a connection is left invariant iff

$$\nabla_{dL_g X}dL_g Y = dL_g (\nabla_{X}Y)$$

For all vector fields $X,Y$ on G, and all $g \in G$.

However, I haven't found any reference for this definition. Could anyone point me to some ressource which defines and discusses left-invariant connections?

Answer 1

The book "Geometry VI - Riemannian geometry" by Postnikov includes a chapter on left-invariant connections. He defines as follows,

wether this is equivalent to the above guess at a definition, I don't know.