  The classification of structurable tori with nontrivial involution, which was
begun by Allison and Yoshii, is completed. New examples of structurable tori
are obtained using a construction of structurable algebras from a semilinear
version of cubic forms satisfying the adjoint identity. The classification uses
techniques borrowed from quadratic forms over Z_2 and from the geometry of
generalized quadrangles. Since structurable tori are the coordinate algebras
for the centreless cores of extended affine Lie algebras of type BC_1, the
results of this paper provide a classification and new examples for this class
of Lie algebras.
