I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance.

Question: I am seeking further exposition of the material in Chapter 4 or 5 of the text. I have prolonged some tableaus, but, I have doubts and would like some additional examples. Can anyone point me to some such exposition on explicit PDEs and Tableaus ?

Let me sketch the goal of the project as it might give insight into which resources we should seek. We have spent some time studying calculus over an associative real algebra. In short, when a function is $\mathcal{A}$-differentiable it must satisfy the $n^2-n$ generalized Cauchy Riemann equations, but, on the level of the algebra variables it's just calculus. On the other hand, we wonder, when can we take a given PDE or system of PDEs and gain insight into the solutions of the system by studying it in terms of the $\mathcal{A}$-calculus. For a basic example, obviously $\mathbb{C}$-variables gives insight into the solutions of Laplace's equation on $\mathbb{R}^2$.

My naive hope was that the machine in Cartan for Beginners could be put to work in ciphering whether a given system of PDEs was compatible with an $\mathcal{A}$-calculus substitution. I hoped there was some natural way to use the Tableaus to judge compatibility. Maybe the hope is misguided.

Added: $\mathcal{A} = \mathbb{R}^n$ paired with an associative multiplication over $\mathbb{R}$. To say $f: \mathcal{A} \rightarrow \mathcal{A}$ is $\mathcal{A}$-differentiable is to say $J_f$ is in the left-regular (matrix) representation. This is equivalent to insisting the algebra multiplication factors out of the differential; $df(v \star w) = df \star w$. For example, $\mathcal{A} = \mathbb{R} \oplus j \mathbb{R}$ with $j^2=1$ gives $$ J_f = \left[ \begin{array}{cc} a & b \\ b & a \end{array}\right]$$ I have a list of examples in: http://link.springer.com/chapter/10.1007/978-1-4614-9332-7_8 and to get a better idea of the PDE idea, see: is this a known method for solving PDEs which illustrates the idea.

Thanks in advance for your help!