We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable structure tensor $$\phi $$ϕ. For the normal case, we prove that a $$\phi $$ϕ-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $$\phi $$ϕ-invariant submanifold $$N$$N everywhere transverse to both the Reeb vector fields but not orthogonal to them, we prove that it is minimal if and only if the angle between the tangential component $$\xi $$ξ (with respect to $$N$$N) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves of $$\xi $$ξ. For the complex case (when just one of the two natural almost complex structures is supposed to be integrable), we prove that a complex submanifold is minimal if and only if it is tangent to both the Reeb vector fields.