Construction of E7(-25) by triality

Let R⁸_V, R⁸₊, R⁸_- be the three 8-dimensional real representations of Spin(8).
The representation of Spin(8) × SL₂(R)_V × SL₂(R)₊ × SL₂ (R)_- on a 56-dimensional real vector space
M = ( R²_V ⊗ R²₊ ⊗ R²_- ) ⊕ ( R⁸_V ⊗ R²_V ) ⊕ ( R⁸₊ ⊗ R²₊ ) ⊕ ( R⁸_- ⊗ R²_- )
can be extended to the representations of G_a = Spin(10, 2) _a × SL₂(R)_a ( a = V, +, - ).

• • Spin(2, 2) ≅ SL₂(R) × SL₂(R).

There are a G_a-invariant symplectic form ω : M × M → R and a G_a-invariant trilinear form τ : M × M × M → R .
We define E_7(-25) = Aut ( M; ω, τ ), which is generated by G_V, G₊, G_-.

• The maximal compact subgroup of E_7(-25) is SO(2)· (E₆/center).
• M appears in the Standard Model in particle physics.