Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators
τ
z
i
(l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators
τ
z
i
and
τ
x
i
associated with l-bits
(
τ
i
)
completely defines the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a nonperturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition
