  Inspired by the combinatorial algebraic approach proposed by Dhurandhar {\it
et al.}, we propose two novel classes of second-generation time-delay
interferometry (TDI) solutions and their further generalization. The primary
strategy of the algorithm is to enumerate specific types of residual laser
frequency noise associated with second-order commutators in products of
time-displacement operators. The derivations are based on analyzing the delay
time residual when expanded in time derivatives of the armlengths order by
order. It is observed that the solutions obtained by such a scheme are
primarily captured by the geometric TDI approach and therefore possess an
intuitive interpretation. Nonetheless, the fully-symmetric Sagnac and
Sagnac-inspired combinations inherit the properties from the original algebraic
approach, and subsequently lie outside of the scope of geometric TDI. We
explicitly show that novel solutions, distinct from existing ones in terms of
both algebraic structure and sensitivity curve, are encountered. Moreover, at
its lowest order, the solution is furnished by commutators of relatively
compact form. Besides the original Michelson-type solution, we elaborate on
other types of solutions such as the Monitor, Beacon, Relay, Sagnac,
fully-symmetric Sagnac, and Sagnac-inspired ones. The average response
functions, residual noise power spectral density, and sensitivity curves are
evaluated for the obtained solutions. Also, the relations between the present
scheme and other existing algorithms are discussed.
