  The application of the Petrov-Galerkin projection method in Cosserat rod
finite element formulations offers significant advantages in simplifying the
expressions within the discrete virtual work functionals. Moreover, it enables
a straight-forward and systematic exchange of the ansatz functions,
specifically for centerline positions and cross-section orientations. In this
concise communication, we present a total Lagrangian finite element formulation
for Cosserat rods that attempts to come up with the least required concepts.
The chosen discretization preserves objectivity and allows for large
displacements/rotations and for large strains. The orientation parametrization
with non-unit quaternions results in a singularity-free formulation.
