  In this paper, we define posets of types B, C, and D. These posets encode the
matrix forms of certain Lie algebras which lie between the algebras of
upper-triangular and diagonal matrices. Interestingly, such type-B, C, and D
Lie poset algebras can be related to Reiner's notion of a parset. Our primary
concern is the index and spectral theories of type-B, C, and D Lie poset
algebras. For an important restricted class, we develop combinatorial index
formulas and, in particular, characterize posets corresponding to Frobenius Lie
algebras. In this latter case we show that the spectrum is binary; that is,
consists of an equal number of 0's and 1's.
