This article investigates the cohomological kernels of field extensions E/F in characteristic two where separable part of E is a quadratic extension and E is quadratic or quartic and purely inseparable over Explicit generators for these kernels are described in both the normal and non-normal cases. The second part of the article gives a construction of an indecomposable division algebra D of index 8 and exponent 2 in characteristic two. The non-normal extensions E/F just considered can arise as maximal subfields of D while in the normal case such a maximal subfield will force a division algebra of exponent 2 to decompose.