Alpha-ketoglutarate-dependent dioxygenase that in vitro catalyzes the regio- and stereoselective hydroxylation of L-lysine, leading to (4R)-4-hydroxy-L-lysine. 2-oxoglutarate + L-lysine + O2 = (4R)-4-hydroxy-L-lysine + CO2 + succinate Binds 1 Fe(2+) ion per subunit. Belongs to the clavaminate synthase family.