I'm reading Gallot-Hulin-Lafontaine, and in section 2.7 they say they following:
I wanted to check that the second $v_g,$ given in a local oriented chart, satisfied the first property. So I took an oriented basis $(e_1,\ldots,e_n)$ satisfying $g(e_i,e_j) = \delta^i_j,$ compared it to the frame $(\partial_1,\ldots,\partial_n)$ (the coordinate vector fields in the $x^i$ coordinates) by writing $e_j = a^i_j \partial_i$ (using summation notation) for some invertible matrix $(a^i_j)$, and computed
$$ v_g(e_1,\ldots,e_n) = v_g(a^i_1 \partial_i, \ldots, a^i_n \partial_i) = \sqrt{\det(g_{i,j})} a^1_1 \cdots a^n_n. $$
This is supposed to equal 1, but I don't see how it follows from the relations
$$\delta^i_j = g(e_i,e_j) = a^k_i a^l_j g_{k,l}.$$
I think I made an error somewhere, but I don't know where. Thanks!