A reduced form for two qubits' density matrix

I know that we can write the density matrix of two qubits in the form
$$
rho = frac14(I + mathbf{a}cdotvec{sigma}otimes I + I otimes mathbf{b}cdotvec{sigma} + t_{ij} , sigma_iotimessigma_j ),
$$
where $mathbf{a}$ and $mathbf{b}$ are vectors and $t_{ij}$ a collection of numbers.

The Question is: We can always write such a density matrix as
$rho = (Uotimes V) (rho_0)(U^daggerotimes V^dagger)$, where $rho_0$ is
$$
rho_0 = frac14(I + mathbf{a}cdotvec{sigma}otimes I + I otimes mathbf{b}cdot vec{sigma} + lambda_i , sigma_iotimessigma_i ).
$$

Notice that in this form we just need 3 terms but in the first representation, we need 9 terms. The unitary transformations seem to reduce this.

Can anyone help me how can I prove this result?