= Solution
The coordinate expression for the <Lie bracket of vector fields> is
$$
[V,W]^\alpha
=V^\mu\partial_\mu W^\alpha-W^\mu\partial_\mu V^\alpha.
$$
For the <Levi-Civita connection>,
$$
V^\mu\nabla_\mu W^\alpha-W^\mu\nabla_\mu V^\alpha
=V^\mu\partial_\mu W^\alpha-W^\mu\partial_\mu V^\alpha
+\Gamma^\alpha_{\mu\nu}(V^\mu W^\nu-W^\mu V^\nu).
$$
The final term vanishes because the <Christoffel symbols> are symmetric in $\mu,\nu$ while the vector product in parentheses is antisymmetric. Hence
$$
\boxed{[V,W]^\alpha
=V^\mu\nabla_\mu W^\alpha-W^\mu\nabla_\mu V^\alpha.}
$$
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