We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $phi$. For the normal case, we prove that a $phi$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $phi$-invariant submanifold $N$ 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 $xi$ (with respect to $N$) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves of $xi$. 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.
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