This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation.
We're working with a finite dimensional real vector space $T$ with basis $\{e_i\}_{i=1}^n$ and its dual space, $T^*$, with basis $\{x^k\}_{k=1}^n$.
First of all we consider the symmetric algebra $S(T^*)$, which can be identified with the algebra of polynomials $\mathbb{R}[x^1,\dots,x^n]$ and is graded as $$ S(T^*)=\bigoplus_{k\in\mathbb{Z}}S^k(T^*), $$ and each $S^k(T^*)$ can be identified with the homogeneous polynomials of degree $k$.
Every derivation $D\in\operatorname{Der}S(T^*)$ is uniquely determined by it's action in $S^1(T^*)$, so there aren't any derivations of degree less than $-1$. Also, there is a natural isomorphism between $T$ and $\operatorname{Der}S(T^*)_{-1}$ given by $$ \partial_v(\eta) = \eta(v), $$ for every $v\in T$ and $\eta\in T^*\cong S^1(T^*)$. With this we obtain the natural identification $$ \partial_{e_i}\leftrightarrow\frac{\partial}{\partial x^i}. $$ With all this it's easily shown that in general, $$ \operatorname{Der}S(T^*)_k\cong\operatorname{Hom}(T^*,S^{k+1}(T^*))\cong S^{k+1}(T^*)\otimes T, $$ and we can identify $S(T^*)\otimes T$ with the set of polynomial vector fields of $T$. With this we can equip $\operatorname{Der}S(T^*)$, seen as a subset of $\operatorname{End}S(T^*)$, with a Lie algebra structure that that preserves the graduation.
Definition Given $h\subset\operatorname{Hom}(T^*,S^{k+1}(T^*))$ in the paper I mention they define the (first) prolongation of $h$ as the set $$ j^{(1)}h:=\{\tau\in\operatorname{Hom}(T^*,h):\tau(u)v=\tau(v)u\,\forall\,u,v\in T^*\}. $$ Now, consider a family of subspaces $g^k\subseteq S^{k+1}(T^*)\otimes T$ for $k=-1,\dots, i$, for some fixed $i$ and with $g^{-1}=T$. What we're going to do is prolong this family of subspaces to form a Lie algebra.
This obscure definition has eluded me for a some days now and I don't know yet how to understand what's really behind it. I've tried to show that my definition is equivalente to Definition 1.1 in this paper, but I no longer think that's true.
Also, according to this paper
A single prolongation of a system of order $q$ consists of augmenting the system with all possible derivatives of its equations [...]
yet I can't make sense of this in my framework.
I tried showing that $$ j^{(1)}g^i\cong\{f\in S^{i+2}(T^*)\otimes T: \partial_w f\in g^i\,\forall\, w\in T\}, $$ but now I'm convinced (not a 100%) this is false.
Edit: This last statement seems to be also used in the book Will Jagy recommends in the comments.
Note: I'm creating a "prolongation-theory" tag which doesn't exist ATM. If the tag is so irrelevant it should be deleted so be it.
Bounty Edit: The jet-bundles tag may be misleading, since I'm interested mostly in the algebraic side of this. I'll leave it as it's a closely related topic.
(probably) Final edit: This question is mostly answered in chapter 4 of the book recommended by James S. Cook in the comments: 'Cartan For Beginners' by Ivey and Landsberg.