A periodic table of (generalised) Grassmannians.
The Vogel plane was introduced by Pierre Vogel in his attempt to build a universal Lie algebra: a framework in which many representation-theoretic invariants (dimensions of irreducible representations, Casimir eigenvalues, …) of a simple Lie algebra become rational functions of three parameters $(\alpha,\beta,\gamma)$, considered up to scaling and permutation.
Given a simple complex Lie algebra $\mathfrak{g}$, decompose the symmetric square of the adjoint representation as $$\operatorname{Sym}^2\mathfrak{g}=\mathbb{C}\oplus Y_2(\alpha)\oplus Y_2(\beta)\oplus Y_2(\gamma)$$ where the three non-trivial summands $Y_2(\alpha),Y_2(\beta),Y_2(\gamma)$ are the irreducible $\mathfrak{g}$-modules characterised by the Casimir eigenvalues attached to $\alpha$, $\beta$ and $\gamma$. The triple $(\alpha:\beta:\gamma)$ is well-defined up to permutation and rescaling, and gives a point of $\mathbb{P}^2/\mathfrak{S}_3$.
In the affine chart drawn here, $\alpha$ has been normalised to $\alpha=-2$ (the squared length of a long root in the usual root-system normalisation). With this choice the sum $$\alpha+\beta+\gamma=h^\vee$$ is the dual Coxeter number of $\mathfrak{g}$, and the Casimir operator acts on the adjoint representation by $2h^\vee$. The horizontal coordinate of the plot is $\beta$ and the vertical coordinate is $\gamma$.
Note that the dual Coxeter number $h^\vee$ generally differs from the Coxeter number $h$ for the non-simply-laced types: e.g. $h^\vee(\mathrm{F}_4)=9$ while $h(\mathrm{F}_4)=12$, and indeed $-2+5+6=9$ at the $\mathrm{F}_4$ point.
The three gold lines contain the classical series:
The two terracotta lines contain the third and fourth rows of the Freudenthal magic square: the exceptional series $\mathfrak{c}_3\subset\mathfrak{a}_5\subset\mathfrak{d}_6\subset\mathfrak{e}_7$ and $\mathfrak{f}_4\subset\mathfrak{e}_6\subset\mathfrak{e}_7\subset\mathfrak{e}_8$.
Most simple Lie algebras have integer coordinates $(\beta,\gamma)$ in this normalisation. The single exception is $\mathfrak{g}_2$, which sits at $(\beta,\gamma)=(10/3,8/3)$—the only point in the picture with non-integer coordinates, and the reason it does not align with any of the other series.
Hover a point to see basic invariants of the associated simple Lie algebra; click a point to go to its Dynkin diagram page.