Understanding the definition of a left-invariant connection on a Lie group Let $\nabla$ be an affine connection on a Lie group G. Here they define $\nabla$ to be left-invariant if, for arbitrary vector fields $X,Y$,
$$d L_g \nabla_X Y = \nabla_{d L_g X}\hspace{0.5mm} d L_g Y \hspace{1mm},$$
where I assume that $d L_g X$ is the differential of $L_g$ acting on a vector field in the following way;  $h \mapsto d(L_g)_h X_h \in T_{gh} G$. 
My first question is about understanding that this definition is well-defined. 


*

*$d L_g X$ is not a vector field, since $h \mapsto d(L_g)_h X_h$ is not in $T_h G$. Therefore $\nabla_{d L_g X}\hspace{0.5mm} d L_g Y$ is not well-defined... or am I reading the expression in a wrong way?
A thought: since $L_g$ is a diffeomorphism, I suppose $d L_g X$ can be identified with the vector field $h \mapsto d({L_g})_{L_g^{-1}(h)}  X_{L_g^{-1}(h)} \in T_h G$. If this is the way to read it, the LHS should be read in the same way.


My second question is about a statement which is supposed to hold, but I don't see why.


*

*Let $\nabla$ be left-invariant. Then it is supposed to hold for left-invariant vector fields X and Y, that $\nabla_X Y$ is itself a left-invariant vector field.


Any solutions, hints or comments would be greatly appreciated!
 A: Hint, but not a complete solution
For a moment, forget about Lie groups. Suppose that $f : M \to N$ is a diffeomorphism of manifolds. Then if you have a tangent vector $v$ at $m \in M$, you can compute $df(m)[v]$ to get a tangent vector at $n = f(m)$. This gives you, for each possible point of $N$, a tangent vector, and that's all that a vector field is. To be more explicit, suppose that we write $h = f^{-1}$, and pick any point $n \in N$. Let $X$ denote a vector field on $M$. Then our new vector field $Y$ on $N$ is defined by 
$$
Y(n) = df(h(n))[X(h(n)],
$$
i.e., we compute the point $m = h(n)$ that's sent to $n$; look at the original vector field there (i.e., $X(h(n))$), and push it forward by the differential of $f$. 
In the case of your Lie group, the manifolds $M$ and $N$ are both $G$, and the map $f$ is $L_q$, and the map $h$ is $L_{g^{-1}}$, but everything else still applies. 
BTW, this whole construction of the "pushforward" of a vector field fails in general if $f$ is not a diffeomorphism, for $f^{-1}(n)$ might consist of multiple points, and pushing forward the vectors from all those points might lead to inconsistent results. 
For your second question, I think you need to 


*

*Write down the definition of what it means for $X$ or $Y$ to be left-invariant. It's a little different from the definition of what it means for $\nabla$ to be left-invariant. 

*Let $Z = \nabla_X Y$. Use your definition from step 1 to say what it means for $Z$ to be left-invariant. That'll involve evaluating $Z(gh)$ and $Z(h)$ and comparing. To express those two things in a way that makes them comparable, you'll use d $L_g \nabla_X Y = \nabla_{d L_g X}\hspace{0.5mm} d L_g Y$,
probably evaluated at the point $h$, so you should write that out as well. And then just simplify with some algebra. I'm pretty sure there's nothing subtle going on here at all -- just definitions of various functions. 
