e7(-5) — noncompact exceptional real form

The real exceptional Lie algebra $\mathfrak{e}_{7(-5)}$ is the noncompact real form of $\mathfrak{e}_{7}(\mathbb C)$ characterized, up to real Lie algebra isomorphism, by a maximal compact subalgebra isomorphic to $\mathfrak{so}(12)\oplus\mathfrak{su}(2)$ . Its real dimension is 133.

Signature convention and magic square

The signature convention is positive minus negative Killing-form directions. In a Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$ it equals $\dim\mathfrak p-\dim\mathfrak k=133-2\cdot69=-5$ . This labels a real form, not its complex dimension. The complexification is the simple complex type $E_7$ algebra.

It occurs among the singly split composition-algebra magic-square outputs. Some sources use the opposite sign for this signature; specifying the maximal compact subalgebra removes that ambiguity.

References

Barton and Sudbery, Magic squares and matrix models of Lie algebras, §3, pp. 11–12, two magic-square tables and maximal compact subalgebras; Theorem 7.2, p. 32.