  We propose an alternative formulation for the exact relations in
three-dimensional homogeneous turbulence using two-point statistics. Our
finding is illustrated with incompressible hydrodynamic, standard and Hall
magnetohydrodynamic turbulence. In this formulation, the cascade rate of an
inviscid invariant of turbulence can be expressed simply in terms of mixed
second-order structure functions. Besides the usual variables like the velocity
${\bf u}$, vorticity $\omega$, magnetic field ${\bf b}$ and the current ${\bf
j}$, the vectors ${\bf u } \times {\boldsymbol \omega}$, ${\bf u} \times {\bf
b}$ and ${\bf j} \times {\bf b}$ are also found to play a key role in the
turbulent cascades. The current methodology offers a simple algebraic form
which is specially interesting to study anisotropic space plasmas like the
solar wind, with in principle a faster statistical convergence than the
classical laws written in terms of third-order correlators.
