Compact Freudenthal magic square

Core idea

The compact Freudenthal magic square assigns a compact real Lie algebra $\mathfrak M(A,B)$ to each ordered pair of real division composition algebras $A,B\in\{\mathbb R,\mathbb C,\mathbb H,\mathbb O\}$, using the Vinberg construction. All outputs below are real; the exceptional symbols mean compact real forms.

Construction and checks

The Tits decomposition relates the entries to $H_3(B)$, and the triality decomposition gives an independent dimension calculation. Transposing the inputs gives isomorphic outputs. The $\mathbb C,\mathbb C$ entry is a genuine direct sum, with dimension $16$.

Octonions and triality

The bottom row contains the exceptional types $F_4,E_6,E_7,E_8$. The $F_4$ entry acts on the Albert algebra, whose frame stabilizer exposes all three eight-dimensional $\operatorname{Spin}(8)$ representations. Compare the decomposition of compact $\mathfrak f_4$ and the different Hom-spaces of those modules.

Real-form convention

$\mathfrak{sp}(3)$ has quaternionic matrix size three and real dimension $21$. The scalar $\mathbb C$ labels a real two-dimensional composition algebra. Split inputs and complexification require separate choices; their objects must not be merged with the compact real ones.

Machine-readable construction

The catalogue stores this table as compact-real-freudenthal: four row IDs, four column IDs, and sixteen cells, each naming its output and construction relationship. The ordered pair is stored explicitly, so a diagram can show both inputs. A construction edge is not a claim that a scalar algebra maps to its output by a homomorphism.

References

1. John C. Baez, “The Octonions,” §4.3, Table 5. Checked section.
2. C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §3 and Theorems 4.3–4.4. Paper.