  Noncommutative pfaffians associated with an orthogonal algebra
$\mathfrak{o}_N$ are some special elements of the universal enveloping algebra
$U(\mathfrak{o}_N)$. Using pfaffians we construct the fourth quantum number
which together with the naturally defined three quantum numbers allow to
classify the states of a five-dimensional quasi-spin. The pfaffians are treated
as creation operators for the new quantum number.
