Title:  Seaweed algebras: Two comprehensive results

In Part One we establish the following strong structural theorem regarding seaweed (biparabolic) algebras.

Theorem:  If g is a Frobenius seaweed algebra of classical type (i.e., type A, B, C, or D), and F is a principal element of g, then the spectrum of ad F consists of an unbroken set of integers centered at one-half-half.  Moreover, the dimensions of the associated eigenspaces forms a symmetric distribution.  The type-D case was very tricky and involved the development of new methods.

In Part Two we provide general closed-form index formulas for seaweed algebras, where the index is given by a polynomial greatest common divisor formulas in the sizes of the parts that define the seaweed.  Using complexity arguments, we show that our list is comprehensive.