Jayaseetha Rau, in her paper Relaxation Phenomena in Spin and Harmonic Oscillator Systems starts from an expression (equation 3.6 in her paper) \begin{equation}\tag{1} U^C(\tau) = \exp(i\lambda\sigma_{1j}\sigma_{2j}\tau) = \exp\left(\frac{i\lambda\tau}{2}[(\sigma_{1j} + \sigma_{2j})^2 - \sigma_{1j}^2 - \sigma_{2j}^2]\right), \end{equation} where $\sigma_{1j}$ and $\sigma_{2j}$ are the Pauli matrices corresponding to two spin systems, to get \begin{equation}\tag{2} U^C(\tau) = \frac{e^{-3i\lambda\tau}}{4}\left[1 + 3e^{4i\tau\lambda} + \sigma_{1j}\sigma_{2j}(e^{4i\tau\lambda} - 1)\right]. \end{equation}
I don't know how to get (2) from (1). Equation (1), in vector form, is just \begin{equation}\tag{3} U^C(\tau) = e^{i\lambda\tau\vec{\sigma}_1\cdot\vec{\sigma}_2} = \exp\left(\frac{i\lambda\tau}{2}[(\vec{\sigma}_{1} + \vec{\sigma}_{2})^2 - \sigma_{1}^2 - \sigma_{2}^2]\right). \end{equation}
Since the square of each Pauli matrix is an identity, \begin{equation}\tag{4} \sigma_1^2 = \sigma_2^2 = 3I_2, \end{equation} $I_2$ being the $2 \times 2$ identity matrix. Therefore, equation (3) becomes \begin{equation}\tag{5} U^C(\tau) = e^{-3i\lambda\tau}\exp\left(\frac{i\lambda\tau}{2}(\vec{\sigma}_{1} + \vec{\sigma}_{2})^2\right). \end{equation}
I am unable to get (2) from (5).