  We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in
seven dimensions. These are $G_2$ structures with $G_2$ being the split real
form of the simple exceptional complex Lie group $G_2$. The first family has
$\tau_2\equiv 0$, while the second family has $\tau_1\equiv\tau_2\equiv 0$. The
families are different in the sense that the first one lives on a homogoneous
space $Sp(2,R)/SL(2,R)_l$, and the second one lives on a homogeneous space
$Sp(2,R)/Sl(2,R)_s$. Here $SL(2,R)_l$ is an $SL(2,R)$ corresponding to the
$\mathfrak{sl}(2,R)$ related to the long roots in the root diagram of
$\mathfrak{sp}(2,R)$, and $SL(2,R)_s$ is an $SL(2,R)$ corresponding to the
$\mathfrak{sl}(2,R)$ related to the short roots in the root diagram of
$\mathfrak{sp}(2,R)$.

Differential Geometry