Periodic matrices in SL(3,Z) will be conjugated to product of periodic matrices in SL(2,Z) by +- indentity on a third integer direction. Is this true?

Sorry, following your comments, maybe something I said is misleading. I state the original question: Consider a periodic automorphism $\phi$ on $Z^3$, can we always find a coordinate on $Z^3$, such that $\phi$ is $\left( \begin{array}{cc} 1 & 0 \\ 0 & A \\ \end{array} \right)$

                        or
                        
                        $\left(
                          \begin{array}{cc}
                            -1 & 0 \\
                            0 & A \\
                          \end{array}
                        \right)$ .