Algebras range in diversity from such fundamental classes as associative and Lie algebras to rather exotic examples such as the Chevalley and Griess algebras. The Chevalley algebra is a commutative nonassociative algebra constructed on the direct sum of the three 8-dimensional representations of the Lie algebra D_4. The Griess algebra is a commutative nonassociative algebra constructed on the direct sum of the trivial representation and the minimal nontrivial representation of the Monster group. Both exceptional algebras have some similarity to the exceptional Lie algebra E_8, but one does not expect to obtain them as substructures of algebras in either general class of algebras. However, it does turn out that these exotic algebras are substructures of examples of a new class of algebras having remarkable similarities to the associative and Lie algebras.