N. L. Gordeev proved that a generalized group identity holds in Chevalley groups with multiply laced root systems. It was also shown that a stronger identity is valid for the Chevalley groups of types B_l and C_l. In the present paper, it is proved that this strong identity is fulfilled in Chevalley groups of type F4 and fails to be true in Chevalley groups of type G₂. The main result of the paper is the last ingredient in the proof of the claim that the lattice of intermediate subgroups between G(F₄,R) and G(F₄,A) is standard for an arbitrary pair of rings R ⊆ A with 2 invertible.