Christoffel symbols and Riemann curvature tensor of a left-invariant metric on a Lie group

Question

Let $G$ be a Lie group equipped with a left-invariant metric, with dimension n. One can write the local coordinates of $G$ as $\phi^a$, whereby $a=1,2..n$. From Milnor's 1976 paper "Curvatures of Left Invariant Metrics on Lie Groups" (equation 5.3) one is able to find the Levi-Civita connection in terms of Lie algebra elements. My question is, how does one extract the Christoffel symbols of the metric, as a function of the coordinates, $\phi^a$? Likewise, I would like to find an expression for the Riemann curvature tensor in terms of the coordinates $\phi^a$, as well.