Let $U \subseteq \mathbb{R}^n$ be an open subset, let $M_n(\mathbb{R})$ be the algebra of real $n \times n$ matrices, and let $B \subseteq M_n(\mathbb{R})$ be a real subalgebra. Assume the coordinates on $\mathbb{R}^n$ are $x_1,\ldots,x_n$. Let $\rho \colon \, U \to B$ be a smooth map, such that the following matrix differential equation holds
$d \rho \wedge \theta = 0.$
where $\theta = (dx_1 \, \ldots \, dx_n)^T$.
For example, if $n=2$, let us say that $B$ consists of all real matrices of the form
$$ B = \left\{ \left( \begin{array}{cc} u & -v \\ v & u \end{array} \right) ; \, u, \, v \in \mathbb{R} \right\}$$
Then the equations $d\rho \wedge \theta = 0$ are equivalent to \begin{align*} du \wedge dx - dv \wedge dy &= 0 \\ dv \wedge dx + du \wedge dy &= 0. \end{align*}
which are equivalent to the Cauchy-Riemann equations.
I was thinking, what does one get if one considered say $n = 4$ and $B$ the real algebra of $4 \times 4$ matrices corresponding to the quaternions, say. Would one get maybe the so called Fueter equations?
$$ B = \left\{ \left( \begin{array}{cccc} u_0 & -u_1 & -u_2 & -u_3 \\ u_1 & u_0 & -u_3 & u_2 \\ u_2 & u_3 & u_0 & -u_1 \\ u_3 & -u_2 & u_1 & u_0 \end{array} \right) ; \, u_0, \, u_1, \, u_2, \, u_3 \in \mathbb{R} \right\} $$
So the equation $d\rho \wedge \theta = 0$ would then become \begin{align*} du_0 \wedge dx_0 - du_1 \wedge dx_1 - du_2 \wedge dx_2 - du_3 \wedge dx_3 &= 0 \\ du_1 \wedge dx_0 + du_0 \wedge dx_1 - du_3 \wedge dx_2 + du_2 \wedge dx_3 &= 0 \\ du_2 \wedge dx_0 + du_3 \wedge dx_1 + du_0 \wedge dx_2 - du_1 \wedge dx_3 &= 0 \\ du_3 \wedge dx_0 - du_2 \wedge dx_1 - du_1 \wedge dx_2 - du_0 \wedge dx_3 &= 0 \end{align*}
So for instance, by looking at the first of these equations, and taking the $dx_0 \wedge dx_1$ component, one gets $$-\frac{\partial u_0}{\partial x_1}-\frac{\partial u_1}{\partial x_0} = 0$$ Similarly, by looking at the second of the previous equations, and taking the $dx_0 \wedge dx_1$ component, one gets $$-\frac{\partial u_1}{\partial x_1}+\frac{\partial u_0}{\partial x_0} = 0.$$ I think the equations $d \rho \wedge \theta = 0$ amount in this case to requiring that the map $$x_0 + i x_1 + j x_2 + k x_3 \mapsto u_0 + i u_1 + j u_2 + k u_3$$ be triholomorphic (which form a special subclass of the class of functions satisfying Fueter's equations).
I guess my post really consists of remarks so far. There are various directions one may go from here. What I would like to know is this. What is the natural setting for these $2$ examples, corresponding to holomorphic functions and triholomorphic functions respectively? There does seem to be a pattern and a general setting. What are other examples of such constructions, and so on?
