Construction of E6(-26) by triality

Let R⁸_V, R⁸₊, R⁸_- be the three 8-dimensional real representations of Spin(8).
The representation of Spin(8) × S(GL₁(R)_V × GL₁(R)₊ × GL₁ (R)_-) on a 27-dimensional real vector space
M = R_V ⊕ R₊ ⊕ R_- ⊕ (R⁸_V ⊗ R_V ) ⊕ ( R⁸₊ ⊗ R₊ ) ⊕ ( R⁸_- ⊗ R_- )
can be extended to the representations of G_a = Spin(9, 1) _a ( a = V, +, - ).
There is a G_a-invariant trilinear form det : M × M × M → R .
We define E_6(-26) = Aut ( M; det ), which is generated by G_V, G₊, G_-.

• The maximal compact subgroup of E_6(-26) is F₄.