  It is known that the universal enveloping algebra of the orthogonal Lie
algebra of size even has a central element expressed in terms of Pfaffian of a
certain matrix alternating along the anti-diagonal (which we call
anti-alternating for short) whose entries are in the univerasal enveloping
algebra. In this paper, we establish minor summation formulae of Pfaffian for
the noncommutative anti-alternating matrix, as well as for commutative
anti-alternating matrix. As an application, we show that the eigenvalues on the
highest weight modules of the central element given by the Pfaffian can be
easily computed.
