Congruence of multilinear forms

LetF:U×⋯×U→K,G:V×⋯×V→Kbe two n-linear forms with n ⩾ 2 on finite dimensional vector spaces U and V   over a field KK. We say that F and G are symmetrically equivalent if there exist linear bijections ϕ1, … , ϕn : U → V such thatF(u1,…,un)=G(ϕi1u1,…,ϕinun)F(u1,…,un)=G(ϕi1u1,…,ϕinun)for all u1, … , un ∈ U and each reordering i1, … , in of 1, … , n. The forms are said to be congruent if ϕ1 = ⋯ = ϕn.Let F and G be symmetrically equivalent. We prove that(i)if K=CK=C, then F and G are congruent;(ii)if K=RK=R, F = F1 ⊕ ⋯ ⊕ Fs ⊕ 0, G = G1 ⊕ ⋯ ⊕ Gr ⊕ 0, and all summands Fi and Gj are nonzero and direct-sum-indecomposable, then s = r and, after a suitable reindexing, Fi is congruent to ±Gi.