The algebraic trace form (as defined by O. Loos) of an element
(x, y) of a (complex) Banach Jordan
pair V, where x or y is in the socle,
is equal to the sum of the products of all spectral values and their
multiplicity. The trace form is calculated for two examples, the
Banach Jordan pair of bounded linear
operators between two Banach spaces, and the Banach Jordan pair
of a quadratic form. Using analytic
multifunctions, it is also shown that the complement of the socle
of a Banach Jordan pair V is either
dense or empty. In the last case, V has finite capacity.