Triality decomposition of compact f4

Statement

For the compact real Lie algebra $\mathfrak f_4$ , the $\operatorname{Spin}(8)$ subgroup fixing a labelled frame in the Albert algebra gives a module decomposition

\mathfrak f_4\cong\mathfrak{so}(8)\oplus8_v\oplus8_+\oplus8_-.

The $\mathfrak{so}(8)$ summand is a Lie subalgebra and has its adjoint action. The other summands are the three inequivalent real eight-dimensional representations linked by triality. This is a decomposition of representations, not of Lie ideals.

Dimensions and the Albert algebra

The dimensions add to $28+8+8+8=52$ . The same three modules occur in the off-diagonal entries of $H_3(\mathbb O)$ , whose decomposition under the frame stabilizer is

H_3(\mathbb O)\cong\mathbb R^3\oplus8_v\oplus8_+\oplus8_-.

These are different representations: the three diagonal lines in the Albert algebra are fixed, while the $28$ -dimensional summand above is adjoint.

References

John C. Baez, “The Octonions,” §4.2, the decomposition into so(8) and its three eight-dimensional modules. Checked section.
John C. Baez, “The Octonions,” §3.4, the triality description of the Albert algebra. Checked section.