  We extend our previous computations for the relative positions of branches of
quaternions to the case of local fields of even characteristic. This is a key
step to understand the set of maximal orders containing a given suborder, which
is useful, for instance, to compute relative spinor images, thus solving the
selectivity problem. In our previous work, the results where given in terms of
the quadratic defect. In the present context, we introduce and characterize an
analogous concept for Artin-Schreier extensions. It is no longer useful to
restrict our attention to orders generated by pure quaternions, as a separable
quadratic extension contains no non-trivial element of null trace. In this work
we state our result for an arbitrary pair of generators, for which we discuss a
more general version of the Hilbert symbol in this context.
