Riemann curvature tensor under covariant derivatives equal to Lie brackets

Question

I'm working on the following question:

Suppose we are given an orthonormal set of vector fields $\lbrace X_1, X_2,\ldots, X_n\rbrace$ on an $n$-dimensional Riemannian manifold $(M, g)$. Suppose further that $$ \nabla_{X_i}X_j = \frac{1}{2}[X_1,X_2],\quad i,j\leq n. $$ Show that the Riemann curvature tensor satisfies $$ R(X_i,X_j,X_k,X_l)=-\frac{1}{4}g([[X_i,X_j],X_k],X_l),\quad i,j,k,l\leq n. $$

And I'm quite stumped. Here's what I know:

1. Since $[X_i,X_j] = \nabla_{X_i}X_j - \nabla_{X_j}X_i$, $\nabla_{X_i}X_j = [X_i,X_j]/2\implies \nabla_{X_i}X_j = - \nabla_{X_j}X_i$.
2. In this case, since we're told that the given set of vector fields are orthonormal, the Koszul formula gives $$ 2g(\nabla_{X_i}X_j,X_k) = -g(X_i,[X_j,X_k]) + g(X_j,[X_k,X_i]) + g(X_k,[X_i,X_j]), $$ and since $g(X_k,[X_i,X_j]) = 2g(\nabla_{X_i}X_j,X_k)$, this reduces to $g(X_i,[X_j,X_k]) = g(X_j,[X_k,X_i])$, so any cyclic permutation of $i$, $j$, and $k$, in the expression $g(X_i,[X_j,X_k])$ leaves the corresponding value of this expression unchanged. That is: $$ g(X_i,[X_j,X_k]) = g(X_k,[X_k,X_i]) = g(X_k,[X_i,X_j]),\quad i,j,k\leq n. $$
3. Since, in general, $[X,Y] = \nabla_XY - \nabla_YX$, we also have that $$ -\nabla_{[X_i,X_j]}X_k = -\nabla_{X_k}[X_i,X_j] - [[X_i,X_j],X_k], $$ from which it follows that $$ R(X_i,X_j,X_k,X_l) = \frac{1}{2}g(\nabla_{X_i}[X_j,X_k] + \nabla_{X_j}[X_k,X_i] - 2\nabla_{X_k}[X_i,X_j],X_l) - g([[X_i,X_j],X_k],X_l), $$
4. The above point leads me to believe that $$ \frac{1}{2}g(\nabla_{X_i}[X_j,X_k] + \nabla_{X_j}[X_k,X_i] - 2\nabla_{X_k}[X_i,X_j],X_l) = -3R(X_i,X_j,X_k,X_l), $$ which, after some fiddling, is equivalent to the assertion that \begin{equation}\label{eq:1} 2\nabla_{X_i}[X_j,X_k]+2\nabla_{X_j}[X_k,X_i]-\nabla_{X_k}[X_i,X_j]-3\nabla_{[X_i,X_j]}X_k = 0 \end{equation} and, to me, this screams Bianchi identity, although evidently I've not been able to make much progress beyond this point. I'm sure the rest of this question has something to do with an explicit evaluation of $\nabla_{X_i}[X_j,X_k]$, but I don't know how to arrive at such an evaluation.
5. As an additional point, it can be shown that the assertion made in the above point can be re-written as $$ R(X_i,X_j)X_k = -\frac{1}{4}[[X_i,X_j],X_k], $$ which would complete the question, but of course we can't assume that the initial assertion holds, as this is what is to be shown.

I'd like to have a go at (what's left of) this question mostly on my own, so if I could be guided in the right direction (as opposed to being given the solution, that is), perhaps with a referral to the facts I'm meant to be relying on in order to show that this relationship holds, for example, that'd be greatly appreciated.

Thanks in advance!

NOTE In the interest of clarity, I want to remark that, unlike what the notation suggests, the set $\lbrace X_1,X_2,\ldots, X_n\rbrace$ is not comprised of coordinate vector fields. Speaking of, I know the notation I’ve used here is quite unique (in comparison to other questions of this nature I've come across on this site, at least); it's how I've been taught to denote things.