Complex sentence: Somewhat confusingly, the tetrad is usually denoted by the coefficients with respect to a coordinate basis e--^ which hides the fact that the choice of a tetrad does not require the additional choice of (local) coordinates x ^\\ mu on the spacetime manifold.
Split into simpler sentences: Somewhat confusingly, the tetrad is usually denoted by the coefficients e--^ with respect to a coordinate basis. This hides the fact that the choice of a tetrad does not actually require the additional choice of (local) coordinates x ^\\ mu on the spacetime manifold.