Using the Cartan-Kahler theory, and results on real algebraic structures, we
prove two embedding theorems. First, the interior of a smooth, compact
3-manifold may be isometrically embedded into a G_2-manifold as an associative
submanifold. Second, the interior of a smooth, compact 4-manifold K, whose
double has a trivial bundle of self-dual 2-forms, may be isometrically embedded
into a Spin(7)-manifold as a Cayley submanifold.