In this section, the Laplacian of any tensor is given as
$\Delta T = g^{ij}(\nabla_i\nabla_j T - \nabla_{\nabla_i e_j}T)$
As I see it, since $\nabla_i{e_j}$ is a vector, we can apply
$\nabla_{V^k e_k}T = V^k \nabla_kT$
and since $\nabla_i{e_j}$ is also the connection,
$\nabla_i{e_j} \equiv \Gamma^k_{ij}e_k$
we end up with
$\nabla_{\nabla_i{e_j}}T = \Gamma^k_{ij}\nabla_k T$
Is that valid? I'm not too confident with differential geometry manipulations, and I didn't see the Laplacian written in coordinate form anywhere. But the rest of it is just coordinate covariant derivatives, which should be more straightforward.