Let F be a field of characteristic 2, Omega(m)(F) the space of m-differential forms over F, d : Omega(m-1 )(F)-> Omega(m)(F) the differential operator, and H-2(m+1) (F) the cokernel of the Artin-Schreier operator P : Omega(m)(F) -> Omega(m)(F)/d Omega(m-1)(F) given on generators by: P(xdx(1)/x(1) boolean AND...boolean AND dx(m)/x(m)) = (x(2)-x) dx(1)/x(1) boolean AND...boolean AND dx(m)/x(m)+ d Omega(m-1)(F). It was shown in [8] that given a multiquadratic purely inseparable extension L/F and a separable quadratic extension K/F, then the kernel of the natural homomorphism H-2(m+1)(F) -> H-2(m+1) (K . L) is equal to the sum of the kernels of the extensions L/F and K/F. In this paper we extend this result to the compositum of a finite purely inseparable multiquadratic extension with a separable biquadratic extension, and we prove that this splitting property of kernels is no longer true by computing the kernel of the compositum of a separable quadratic (or biquadratic) extension with a simple purely inseparable extension of arbitrary degree. (C) 2019 Elsevier Inc. All rights reserved.