  We propose polynomial-time algorithms for finding nontrivial zeros of
quadratic forms with four variables over rational function fields of
characteristic 2. We apply these results to find prescribed quadratic subfields
of quaternion division division algebras and zero divisors in $M_2(D)$, the
full matrix algebra over a division algebra, given by structure constants. We
also provide an implementation of our results in MAGMA which shows that the
algorithms are truly practical.
