Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)?
I am especially interested in the quaternary case. There exist some algorithms for ternary quadratic forms, some of which are implemented in Magma.
The only algorithm for quaternary forms over the rationals I know of is described in Algorithms for solving rational quadratics (Schicho, Pílniková). It works by dividing the equation up into two appropriate ternary forms. However I see no easy way to make it work over number fields.