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We find two different families of    $$\mathbf{Sp}(4,\mathbb{R})$$  Sp ( 4 , R ) symmetric    $$G_2$$  G 2 structures in seven dimensions. These are    $$G_2$$  G 2 structures with    $$G_2$$  G 2 being the split real form of the simple exceptional complex Lie group    $$G_2$$  G 2 . The first family has    $$\tau _2\equiv 0$$  τ 2 ≡ 0 , while the second family has    $$\tau _1\equiv \tau _2\equiv 0$$  τ 1 ≡ τ 2 ≡ 0 , where    $$\tau _1$$  τ 1 ,    $$\tau _2$$  τ 2 are the celebrated    $$G_2$$  G 2 -invariant parts of the intrinsic torsion of the    $$G_2$$  G 2 structure. The families are different in the sense that the first one lives on a homogeneous space    $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_l$$  Sp ( 4 , R ) / SL ( 2 , R ) l , and the second one lives on a homogeneous space    $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s$$  Sp ( 4 , R ) / SL ( 2 , R ) s . Here    $$\mathbf{SL}(2,\mathbb{R})_l$$  SL ( 2 , R ) l is an    $$\mathbf{SL}(2,\mathbb{R})$$  SL ( 2 , R ) corresponding to the    $$\mathfrak{sl}(2,\mathbb{R})$$  sl ( 2 , R ) related to the long roots in the root diagram of    $$\mathfrak{sp}(4,\mathbb{R})$$  sp ( 4 , R ) , and    $$\mathbf{SL}(2,\mathbb{R})_s$$  SL ( 2 , R ) s is an    $$\mathbf{SL}(2,\mathbb{R})$$  SL ( 2 , R ) corresponding to the    $$\mathfrak{sl}(2,\mathbb{R})$$  sl ( 2 , R ) related to the short roots in the root diagram of    $$\mathfrak{sp}(4,\mathbb{R})$$  sp ( 4 , R ) .

On certain classes of $$\mathbf{Sp}(4,\mathbb{R})$$ symmetric $$G_2$$ structures