  We give a construction of the compact real form of the Lie algebra of type
$E_6$, using the finite irreducible subgroup of shape
$3^{3+3}:\mathrm{SL}_3(3)$, which is isomorphic to a maximal subgroup of the
orthogonal group $\Omega_7(3)$. In particular we show that the algebra is
uniquely determined by this subgroup. Conversely, we prove from first
principles that the algebra satisfies the Jacobi identity, and thus give an
elementary proof of existence of a Lie algebra of type $E_6$. The compact real
form of $F_4$ is exhibited as a subalgebra.
