Below is a picture of an excerpt of Lavendhomme's book on synthetic differential geometry.
I'm confused about the "structure constants" $\gamma_{ij}^k$. Initially I thought the superscript $k$ is just an index and not an exponent, and that these are the coordinate vectors of the products $e_i\cdot e_j$. Now I'm no longer certain. Which is it?
It's an index. The reason it's a superscript is to indicate that it has a different variance from the subscripts.