I am studying quantum computation and have been stuck in a simple question that may look silly to experts of mathematics. Every two-qubit operation is expressed as an element of $SU(4)$. All the textbooks and papers in physics I saw state that every element $U \in SU(4)$ can be written of the form
$$U = \exp \left( -i \sum_{j, j'=0,x,y,z, (j,j')\ne(0,0)} t_{j j'} \sigma_j \otimes \sigma_{j'} \right)$$
with real parameters $t_{j j'}$ for the Pauli matrices $\sigma_j \, (\sigma_0 \equiv I_2)$.
I am wondering if it is true for $U = \pm i I_4 \in SU(4)$, which make me feel no parameter $t_{j j'}$ gives them.
I understand I misunderstand something... but I do not know what the correct theory is.
In addition, I want to know how to determine for given 16 components of an $SU(4)$ matrix whether it can be written in the exp form or not.
Note that $\pm i I_4 = U(\pm \frac{\pi}{2}),$ where $$ U(t) = \begin{pmatrix} e^{it} & 0 & 0 & 0 \\ 0 & e^{it} & 0 & 0 \\ 0 & 0 & e^{it} & 0 \\ 0 & 0 & 0 & e^{-3it} \\ \end{pmatrix} \in SU(4). $$
Now, $$ U'(0) = \begin{pmatrix} i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & 0 & 0 & -3i \\ \end{pmatrix} = i (\sigma_0 \otimes \sigma_3 + \sigma_3 \otimes \sigma_0 - \sigma_3 \otimes \sigma_3) =: i\Sigma , $$ i.e. $U(t) = \exp(it\Sigma).$