The spin modules for the Lie algebras of type Dn and Bn are constructed through minuscule systems. It is shown that the unique (up to multiplication by a nonzero scalar) g-invariant bilinear form on the modules is nondegenerate. By deﬁning an equation for the height of weights of the modules, it is shown that the bilinear form is symplectic when n = 1 or 2(mod 4), and orthogonal when n = 0 or 3(mod 4).
Lonzinsky, Sergey, "Properties of the g-Invariant Bilinear Form on the Spin Representations of the Simple Lie Algebras of Type Dn and Bn" (2013). Undergraduate Honors Theses. 428.