  We present an efficient method for construction of a fully anti-commutative
set of Pauli generators (elements of the Pauli group) from a commutative set of
operators that are composed exclusively from Pauli $\hat x_i$ operators (purely
X generators) and sorted by an associated numerical measure, such as absolute
energy gradients. Our approach uses the Gauss-Jordan elimination applied to a
binary matrix that encodes the set of X generators to bring it to the reduced
row echelon form, followed by the construction of an anti-commutative system in
a standard basis by means of a modified Jordan-Wigner transformation and
returning to the original basis. The algorithm complexity is linear in the size
of the X set and quadratic in the number of qubits. The resulting
anti-commutative sets are used to construct the qubit coupled cluster Ansatz
with involutory linear combinations of anti-commuting Paulis (QCC-ILCAP)
proposed in [J. Chem. Theory Comput. 2021, 17, 1, 66-78]. We applied the
iterative qubit coupled cluster method with the QCC-ILCAP Ansatz to
calculations of ground-state potential energy curves for symmetric stretching
of the water molecule (36 qubits) and dissociation of N$_2$ (56 qubits).
