Relationship between Riemannian Exponential Map and Lie Exponential Map

It is well known that for a matrix Lie group the Lie exponential map is $e ^z$. This maps a tangent vector $z$ at the identity to a group element.

On the other hand the general Riemannian exponential map centered at point $x$ is given by $\exp _x \triangle$ which maps a tangent vector $\triangle$ at point $x$ (not necessarily identity element) to a group element.

Is there a relationship between these two exponential maps?

For example is below formula correct? If so, are there any conditions involved?

$\exp _x \triangle = xe ^{x^{-1}\triangle}$

Notice that you need to pick a metric on a Lie group for the "general Riemannian exponential map" to be defined.

If you happen to pick an invariant metric on a Lie group, then every geodesic is (locally) a translate of a 1-parameter subgroup (so essentially both exponentials are the same thing)

I don't know what happens if there are no invariant metrics.