Real-form conventions for magic squares

Core idea

The compact Freudenthal magic square uses the real division composition algebras $\mathbb R,\mathbb C,\mathbb H,\mathbb O$ and returns compact real Lie algebras. Here $\mathbb C$ is a two-dimensional real input algebra, not the base field of the construction.

Changing the input form

The split complex numbers, split quaternions and split octonions are separate real composition algebras. Their quadratic forms are indefinite, and they have zero divisors. Inserting them into the same construction changes the real forms of its outputs. Using split inputs along only one axis need not give the split form of every output.

Changing the scalar field

Complexifying a compact real output produces a complex Lie algebra. This is different from viewing that real algebra as a complex vector space: a scalar extension is required. The compact and split real forms of a given complex type can have isomorphic complexifications without being isomorphic over $\mathbb R$.

Catalogue convention

The machine table is named compact-real-freudenthal; each axis references the actual input object IDs, and each cell references a real output object. Split scalar objects and complex exceptional objects have their own records. A missing split-square cell means the construction is not yet catalogued in that table, not that no such construction exists. The sedenions are not composition algebras and are not a fifth input to this square.

References

1. C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §§2–3, especially pp. 4 and 10–12, division versus split composition algebras and the real magic-square tables. Paper.