The pair of elements that form both oxides and nitrides, when burnt in air are

Reaction of magnesium or calcium with oxygen gives magnesium oxide or calcium oxide.

$2\mathrm{Mg}+\underset{\mathrm{From} air}{{\mathrm{O}}_2}⟶2\mathrm{MgO}$

Reaction of magnesium or calcium with nitrogen gives ${\mathrm{Mg}}_3{\mathrm{N}}_2 \mathrm{or} {\mathrm{Ca}}_3{\mathrm{N}}_2$. The chemical reaction  is as follows.
$3\mathrm{Mg}+\underset{\mathrm{From} air}{{\mathrm{N}}_2}⟶{\mathrm{Mg}}_3{\mathrm{N}}_2$

Hence, metal are magnesium and calcium.