  Let $\C$ be a genus 2 curve defined over $k$, $char (k) =0$. If $\C$ has a
$(3,3)$-split Jacobian then we show that the automorphism group $Aut(\C)$ is
isomorphic to one of the following: $\bZ_2, V_4, D_8$, or $D_{12}$. There are
exactly six $\bC$-isomorphism classes of genus two curves $\C$ with $Aut(\C)$
isomorphic to $D_8$ (resp., $D_{12}$). %We compute their absolute invariants
$i_1, i_2, i_3$. We show that exactly four (resp., three) of these classes with
group $D_8$ (resp., $D_{12}$) have representatives defined over $\bQ$. We
discuss some of these curves in detail and find their rational points.
