I am considering a tensor (in particular, the electric field), defined by $$E_m = g_{ij}^k c_{k\ell}^{ij}S_{\ell m} $$ Ultimately, this means that the tensor E is a rank 1, type (0,1) tensor, containing one vector, as a map $\mathbb{R}^3\to \mathbb{R}, \mathbf{v}\mapsto \mathbf{E}(\mathbf{v})\in\mathbb{R}$. After performing the multiplication of the matrix representations of each of the tensors above, we have $$ g_1^4 c_{12}^{12} \begin{bmatrix} S_4 \\ S_5 \\ S_6 \end{bmatrix} = \begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix}=[\mathbf{E}]_{[\hat{x},\hat{y},\hat{z}]}$$ As can be seen, the electric field tensor is in the standard Cartesian basis (unsure if I should be using the dual basis here). My goal is to find a change-of-basis matrix $A$ into the basis $\{\hat{\mathbf{e}}_1,\hat{\mathbf{e}}_2,\hat{\mathbf{e}}_3 \}$, with the only condition on this new basis being that one of the $\hat{\mathbf{e}}_i$ must be the vector $\langle 1,1,1 \rangle$. Secondly, I want the matrix representation of the electric field in the new basis to be of the form $$\begin{bmatrix}E_1 \\ 0 \\ 0 \end{bmatrix} $$ or something similar, with the one nonzero element being that which corresponds to the $\langle 1,1,1\rangle $ vector.
The transformation law for a a rank one tensor, such as this one, as far as I know is $E'_n = A_n^m E_m$. In other words, this tensor $A_n^m$ is a matrix, but the complication arises with the fact that I want the matrix of $E'$ to be also 3x1, and because I don't have an exact basis I am transforming to, just a family of such bases. Note also that these bases are defined on a hexagonal plane, which is why I made no demand of orthogonality in the new basis. I would appreciate tips on how to proceed with such a problem.