My rep. is not high enough to leave a comment so this has to go here. I comment that I have done a detailed study of matrices similar to those you mention for cubic fields only. A small part of that will be in: A cubic generalization of Brahmagupta's identity, J. Ramanujan Math. Soc., 32, (2017) no. 4, 327-337. Also, see : Parametrization of 2-dimensional torus, which is related. bf Edit 29/11/17: I can give a partial answer to the question in the following: Let $$N_{\mathcal{C}}^{(\alpha )} = \left( \begin{array}{ccc} u & -a d y & -a d x-b d y \\ x & u-b x-c y & -c x-d y \\ y & a x & u-c y \\ \end{array} \right) ,$$ where $\mathcal{C}(x, y) = (a, b, c, d)$ is an index form for the cubic field $K = \mathbb{Q}(\delta )$, $\delta \in \mathbb{R} $ satisfies $\mathcal{C}(\delta , 1) = 0$, and $u,x,y$ are coefficients of Belabas' integral basis $\{ 1, \theta , \phi \}$, $\theta = a \delta $, $\phi = a \delta^2 + b \delta $,and $\alpha = u + x \theta + y \phi $. With $\alpha \in K$ or the ring of integers, the matrices commute with one another, their determinant is the norm, their sum can be used to produce the sum $\alpha_1 + \alpha_2$, the trace of the matrix is the trace of $\alpha $, the inverse of the matrix can be used to produce the inverse of the $\alpha \in K$. One extracts the first column and builds $u + x \theta + y \phi \in K$ from it. Now the proofs of all of this should work if instead of letting $u,x,y \in \mathbb{Q}$, we let $u, x, y \in E$, an extension of $K$. From that it should be possible to find matrices that do the same job as these do for the appropriate extension field.

My rep. is not high enough to leave a comment so this has to go here. I comment that I have done a detailed study of matrices similar to those you mention for cubic fields only. A small part of that will be in: A cubic generalization of Brahmagupta's identity, J. Ramanujan Math. Soc., 32, (2017) no. 4, 327-337. Also, see : Parametrization of 2-dimensional torus, which is related. bf Edit 29/11/17: I can give a partial answer to the question in the following: Let $$N_{\mathcal{C}}^{(\alpha )} = \left( \begin{array}{ccc} u & -a d y & -a d x-b d y \\ x & u-b x-c y & -c x-d y \\ y & a x & u-c y \\ \end{array} \right) ,$$ where $\mathcal{C}(x, y) = (a, b, c, d)$ is an index form for the cubic field $K = \mathbb{Q}(\delta )$, $\delta \in \mathbb{R} $ satisfies $\mathcal{C}(\delta , 1) = 0$, and $u,x,y$ are coefficients of Belabas' integral basis $\{ 1, \theta , \phi \}$, $\theta = a \delta $, $\phi = a \delta^2 + b \delta $,and $\alpha = u + x \theta + y \phi $. With $\alpha \in K$ or the ring of integers, the matrices commute with one another, their determinant is the norm, their sum can be used to produce the sum $\alpha_1 + \alpha_2$, the trace of the matrix is the trace of $\alpha $, the inverse of the matrix can be used to produce the inverse of the $\alpha \in K$. One extracts the first column and builds $u + x \theta + y \phi \in K$ from it. Now the proofs of all of this should work if instead of letting $u,x,y \in \mathbb{Q}$, we let $u, x, y \in F$, another field for which we know the associated matrices, e.g. the quadratic field $\mathbb{Q}(\sqrt{D})$. From that it should be possible to find matrices that do the same job as these do for the appropriate extension field, e.g. the normal closure of $K$.

My rep. is not high enough to leave a comment so this has to go here. I comment that I have done a detailed study of matrices similar to those you mention for cubic fields only. A small part of that will be in: A cubic generalization of Brahmagupta's identity, J. Ramanujan Math. Soc., 32, (2017) no. 4, 327-337. Also, see : Parametrization of 2-dimensional torus, which is related.

My rep. is not high enough to leave a comment so this has to go here. I comment that I have done a detailed study of matrices similar to those you mention for cubic fields only. A small part of that will be in: A cubic generalization of Brahmagupta's identity, J. Ramanujan Math. Soc., 32, (2017) no. 4, 327-337. Also, see : Parametrization of 2-dimensional torus, which is related. bf Edit 29/11/17: I can give a partial answer to the question in the following: Let $$N_{\mathcal{C}}^{(\alpha )} = \left( \begin{array}{ccc} u & -a d y & -a d x-b d y \\ x & u-b x-c y & -c x-d y \\ y & a x & u-c y \\ \end{array} \right) ,$$ where $\mathcal{C}(x, y) = (a, b, c, d)$ is an index form for the cubic field $K = \mathbb{Q}(\delta )$, $\delta \in \mathbb{R} $ satisfies $\mathcal{C}(\delta , 1) = 0$, and $u,x,y$ are coefficients of Belabas' integral basis $\{ 1, \theta , \phi \}$, $\theta = a \delta $, $\phi = a \delta^2 + b \delta $,and $\alpha = u + x \theta + y \phi $. With $\alpha \in K$ or the ring of integers, the matrices commute with one another, their determinant is the norm, their sum can be used to produce the sum $\alpha_1 + \alpha_2$, the trace of the matrix is the trace of $\alpha $, the inverse of the matrix can be used to produce the inverse of the $\alpha \in K$. One extracts the first column and builds $u + x \theta + y \phi \in K$ from it. Now the proofs of all of this should work if instead of letting $u,x,y \in \mathbb{Q}$, we let $u, x, y \in E$, an extension of $K$. From that it should be possible to find matrices that do the same job as these do for the appropriate extension field.