  An ideal is a classical object of study in the field of algebraic number
theory. In maximal quadratic orders of number fields, ideals usually
represented by the $\mathbb Z$-basis. This form of representation is used in
most of the algorithms for ideal manipulation. However, this is not the only
option, as an ideal can also be represented by its generating set. Moreover, in
certain quadratic orders of the ring of integral quaternions, every ideal can
be represented by a single quaternion, called a pseudo generator.
  In this paper, we develop algorithms which allow us to manipulate ideals
using solely their pseudo generators. We also demonstrate the connection
between the three squares problem and factorization. This result comes as an
extension of a number factoring technique discovered by Fermat, which allows
the factoring of an integer if a pair of its two squares representations is
known. In addition to this, we present a number of identities, which ideals
must satisfy; some computational data on the number of ambiguous classes,
generated by the equation $\rho\mu = -\mu\rho$; and a new approach to the
problem of finding a non-trivial divisor of a class number.
