On page 155 of Vector and Tensor Analysis with Applications, by A.I Borishenko and I.E. Tarapov, the authors assert that,
$$\frac{\partial {\bf e}_j}{\partial x^k} = \left\{ i \atop j \; k \right\} {\bf e}_i$$ and $$ [i,jk] = \frac{\partial {\bf e}_j}{\partial x_k} {\bf e}^i $$ imply that $$[i , j k] = {\bf e}_i \cdot \frac{\partial {\bf e}_j}{\partial x^k}$$
Unfortunately, I am unable to show how the final equation follows from the first two.
Update
Extract from the text
The derivative of ${\bf e}_j$ is first covariant, then contravariant, then covariant. Besides surely $\left\{ i \atop j \; k \right\}$ cannot be expansion coefficients of $\frac{\partial {\bf e}_j}{\partial x_k}$. Could this be a typo?
An answer to this question has been posted on mathoverflow.
I have added this extra content to stop the system from converting my answer to a comment.