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This paper concerns an associative graded algebra A that is the envelopingalgebra of a Lie algebra with exponential growth. The algebra A is3-Calabi-Yau. There is a Z-form of A so for every field k there is an algebraA_k. An algebraic group of type G_2 acts as degree-preserving automorphisms ofA.  The algebra A is generated by 7 elements modulo 7 homogeneous quadraticrelations. It can be constructed from the octonions; the same constructionapplied to the quaternions produces the commutative polynomial ring in 3variables.  If V is the 7-dimensional irreducible representation of the complexsemisimple Lie algebra of type G_2, then A is isomorphic to the tensor algebraTV modulo the ideal generated by the submodule of V \otimes V that isisomorphic to V.  A can be defined as a superpotential algebra derived from a 3-form on R^7having an open GL(7) orbit and compact isotropy group.  Classification of the finite-dimensional representations of A is equivalentto classifying square matrices Y with purely imaginary octonion entries suchthat the imaginary part of Y^2 is zero.  We show that A is a Koszul algebra. Its Koszul dual is a quotient of theexterior algebra on 7 variables, has Hilbert series 1+7t+7t^2+t^3, and is asymmetric Frobenius algebra.  Although not noetherian, A is graded coherent (proved by Piontkovski).

A 3-Calabi-Yau algebra with G_2 symmetry constructed from the octonions