Since you're mentioning the Jordan decomposition and Lie algebras, the connection between the representation theory of $\mathfrak{sl}_2$ (explicitly: Jacobson-Morozov, classification of irreducible representations, primitives), the Jordan decomposition of nilpotent operators and the weight filtration associated with a nilpotent operator comes to my mind. To which extent this is suitable heavily depends on the audience's background in $\mathfrak{sl}_2$-representations and linear algebra. Disclaimer: quite a few people I know consider this useless/ridiculous overkill. Moreover, I can't offer any combinatorics here and the representation theory is not at all deep or even very interesting. Nonetheless, I like it a lot, so here it goes:
The central concept is the weight filtration associated with a nilpotent endomorphism $N\colon V\to V$ of a finite-dimensional $k$-vector space $V$, $W_{\bullet} = W_{\bullet}(N)$:
$$0 = W_{-n-1}\subset W_{-n}\subset\dots\subset W_{n-1}\subset W_n = V$$
which is, up to isomorphism, uniquely determined by the following two properties:
- $N$ decreases the degree by two, i.e., for each integer $i$, $NW_{i}\subset W_{i-2}$, and
- for each integer $i$, the map $N^i\colon W_i\to W_{-i}$ induces an isomorphism $\overline{N}^i\colon \mathrm{Gr}_i(W_{\bullet})\to\mathrm{Gr}_{-i}(W_{\bullet})$.
This is a classical thing in Hodge theory; the reference I'd like to mention here is Deligne, La Conjecture de Weil. II, IHES 1980, (1.6) Autour de Jacobson-Morosov. A web search will find you some more references.
Example 1 (Relation with the Jordan normal form) — Let $N$ have a Jordan decomposition with basis given by elements $v_1,v_2,\dots,v_k$ and their images under $N$, $N^2$, etc. Then we get a splitting of the filtration as follows:
Define the primitive pieces $P_i\subset W_i$ as the span of those $v_j$ such that $N^{i}v_j\not=0$, but $N^{i+1}v_j=0$. Then $\mathrm{Gr}_i(W_{\bullet}) \cong P_i\oplus NP_{i+2}\oplus N^2P_{i+4}\oplus \dots$ via the obvious maps.
Abstractly, however, the primitive pieces as subspaces of the graded pieces are independent of the chosen basis, as $P_i = \ker(\overline N^{i+1}\colon \mathrm{Gr}_i(W_{\bullet})\to\mathrm{Gr}_{-i-2}(W_{\bullet}))$.
Thus, reading the above backwards, the Jordan normal form itself, without the basis, is already encoded in the (dimensions of the) primitive pieces.
Moreover, basic information about the normal form can be read off the filtration without looking at the primitive pieces.
Choosing a Jordan basis usually (with the standard algorithm) means choosing some a particular vectors, calculating their images under $N$, repeatedly, and then making some further choices depending on the prior choices, and so on.
But if you know the weight filtration, choosing a Jordan basis boils down to the (conceptually easier) choice of a splitting and then simply choosing bases of the primitive pieces.
Example 2 (Relation with $\mathfrak{sl}_2$) — By the Jacobson-Morozov Theorem, $N$ is part of an $\mathfrak{sl}_2$-triple $(M,H,N)$. Here, $H$ determines (and is determined by) a splitting of the filtration, since the graded pieces are the eigenspaces of $H$. Moreover, from the grading and $N$, one can inductively construct $M$. (This yields a proof of Jacobson-Morozov.) Moreover, there is an obvious relation between the highest weight vectors (or the decomposition into irreducible representations) and the primitive pieces.
A nice concrete example is the exterior algebra of a complex symplectic vector space with nilpotent endomorphism given by wedge product with the fundamental form (the Lefschetz operator). But to get the representation one usually considers, one should better take the dual Lefschetz operator, which is given by contraction with the fundamental form. In either case, a natural splitting is given by the degree and the stuff about primitive parts (“Lefschetz decomposition”) is of great importance in the (local) theory of Kähler manifolds.
Let me conclude with a remark: One can spend quite some time talking about the weight filtration and the Jordan normal form without ever mentioning $\mathfrak{sl}_2$. Perhaps it is an option to solely mention that there is some $\mathfrak{sl}_2$-representation theory lurking in the background. But that would feel quite unsatisfactory, I’m afraid.
