This graph is embeddable.
It is unknown whether this graph is grid embeddable.

Embeddability of the graph reduces to the truth of a formula over the reals,
which can be decided with the
reduce algebra package
using the following script.

load_package redlog;
rlset R;
procedure d(x,y);
(first x) * (first y) +
(second x) * (second y) +
(third x) * (third y);
procedure k(x,y);
{(second x)*(third y) - (third x)*(second y),
(third x)*(first y) - (first x)*(third y),
(first x)*(second y) - (second x)*(first y)};
v0c1 := 1; v0c2 := 0; v0c3 := 0;
v1c1 := 0; v1c2 := 1; v1c3 := 0;
v0 := {v0c1, v0c2, v0c3};
v1 := {v1c1, v1c2, v1c3};
v2 := {v2c1, v2c2, v2c3};
neq0 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),v2);
neq1 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),k(k(v2,v1),v1));
neq2 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),k(k(k(v2,v1),v2),k(v0,v1)));
neq3 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),k(k(k(v2,v1),v2),v0));
neq4 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),v0);
neq5 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)));
neq6 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),k(v2,k(v0,v1)));
neq7 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),v1);
neq8 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),k(k(v2,v1),v2));
neq9 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1)),k(v0,v1));
neq10 := k(v2,k(k(v2,v1),v1));
neq11 := k(v2,k(k(k(v2,v1),v2),k(v0,v1)));
neq12 := k(v2,k(k(k(v2,v1),v2),v0));
neq13 := k(v2,v0);
neq14 := k(v2,k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))));
neq15 := k(v2,k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)));
neq16 := k(v2,v1);
neq17 := k(v2,k(v0,v1));
neq18 := k(v2,k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq19 := k(k(k(v2,v1),v1),k(k(k(v2,v1),v2),k(v0,v1)));
neq20 := k(k(k(v2,v1),v1),k(k(k(v2,v1),v2),v0));
neq21 := k(k(k(v2,v1),v1),v0);
neq22 := k(k(k(v2,v1),v1),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))));
neq23 := k(k(k(v2,v1),v1),k(v2,k(v0,v1)));
neq24 := k(k(k(v2,v1),v1),k(k(v2,v1),v2));
neq25 := k(k(k(v2,v1),v1),k(v0,v1));
neq26 := k(k(k(v2,v1),v1),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq27 := k(k(k(k(v2,v1),v2),k(v0,v1)),k(k(k(v2,v1),v2),v0));
neq28 := k(k(k(k(v2,v1),v2),k(v0,v1)),v0);
neq29 := k(k(k(k(v2,v1),v2),k(v0,v1)),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))));
neq30 := k(k(k(k(v2,v1),v2),k(v0,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)));
neq31 := k(k(k(k(v2,v1),v2),k(v0,v1)),k(v2,k(v0,v1)));
neq32 := k(k(k(k(v2,v1),v2),k(v0,v1)),v1);
neq33 := k(k(k(k(v2,v1),v2),k(v0,v1)),k(v2,v1));
neq34 := k(k(k(k(v2,v1),v2),v0),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))));
neq35 := k(k(k(k(v2,v1),v2),v0),k(v2,k(v0,v1)));
neq36 := k(k(k(k(v2,v1),v2),v0),v1);
neq37 := k(k(k(k(v2,v1),v2),v0),k(v0,v1));
neq38 := k(k(k(k(v2,v1),v2),v0),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq39 := k(k(k(k(v2,v1),v2),v0),k(v2,v1));
neq40 := k(v0,k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))));
neq41 := k(v0,k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)));
neq42 := k(v0,k(v2,k(v0,v1)));
neq43 := k(v0,k(k(v2,v1),v2));
neq44 := k(v0,k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq45 := k(v0,k(v2,v1));
neq46 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)));
neq47 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))),v1);
neq48 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))),k(k(v2,v1),v2));
neq49 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))),k(v0,v1));
neq50 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))),k(v2,v1));
neq51 := k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(v2,k(v0,v1)));
neq52 := k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v1);
neq53 := k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(v2,v1),v2));
neq54 := k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(v0,v1));
neq55 := k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(v2,v1));
neq56 := k(k(v2,k(v0,v1)),v1);
neq57 := k(k(v2,k(v0,v1)),k(k(v2,v1),v2));
neq58 := k(k(v2,k(v0,v1)),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq59 := k(k(v2,k(v0,v1)),k(v2,v1));
neq60 := k(v1,k(k(v2,v1),v2));
neq61 := k(v1,k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq62 := k(k(k(v2,v1),v2),k(v0,v1));
neq63 := k(k(k(v2,v1),v2),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq64 := k(k(v0,v1),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))));
neq65 := k(k(v0,v1),k(v2,v1));
neq66 := k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1));
phi :=
(first neq0 neq 0 or
second neq0 neq 0 or
third neq0 neq 0) and
(first neq1 neq 0 or
second neq1 neq 0 or
third neq1 neq 0) and
(first neq2 neq 0 or
second neq2 neq 0 or
third neq2 neq 0) and
(first neq3 neq 0 or
second neq3 neq 0 or
third neq3 neq 0) and
(first neq4 neq 0 or
second neq4 neq 0 or
third neq4 neq 0) and
(first neq5 neq 0 or
second neq5 neq 0 or
third neq5 neq 0) and
(first neq6 neq 0 or
second neq6 neq 0 or
third neq6 neq 0) and
(first neq7 neq 0 or
second neq7 neq 0 or
third neq7 neq 0) and
(first neq8 neq 0 or
second neq8 neq 0 or
third neq8 neq 0) and
(first neq9 neq 0 or
second neq9 neq 0 or
third neq9 neq 0) and
(first neq10 neq 0 or
second neq10 neq 0 or
third neq10 neq 0) and
(first neq11 neq 0 or
second neq11 neq 0 or
third neq11 neq 0) and
(first neq12 neq 0 or
second neq12 neq 0 or
third neq12 neq 0) and
(first neq13 neq 0 or
second neq13 neq 0 or
third neq13 neq 0) and
(first neq14 neq 0 or
second neq14 neq 0 or
third neq14 neq 0) and
(first neq15 neq 0 or
second neq15 neq 0 or
third neq15 neq 0) and
(first neq16 neq 0 or
second neq16 neq 0 or
third neq16 neq 0) and
(first neq17 neq 0 or
second neq17 neq 0 or
third neq17 neq 0) and
(first neq18 neq 0 or
second neq18 neq 0 or
third neq18 neq 0) and
(first neq19 neq 0 or
second neq19 neq 0 or
third neq19 neq 0) and
(first neq20 neq 0 or
second neq20 neq 0 or
third neq20 neq 0) and
(first neq21 neq 0 or
second neq21 neq 0 or
third neq21 neq 0) and
(first neq22 neq 0 or
second neq22 neq 0 or
third neq22 neq 0) and
(first neq23 neq 0 or
second neq23 neq 0 or
third neq23 neq 0) and
(first neq24 neq 0 or
second neq24 neq 0 or
third neq24 neq 0) and
(first neq25 neq 0 or
second neq25 neq 0 or
third neq25 neq 0) and
(first neq26 neq 0 or
second neq26 neq 0 or
third neq26 neq 0) and
(first neq27 neq 0 or
second neq27 neq 0 or
third neq27 neq 0) and
(first neq28 neq 0 or
second neq28 neq 0 or
third neq28 neq 0) and
(first neq29 neq 0 or
second neq29 neq 0 or
third neq29 neq 0) and
(first neq30 neq 0 or
second neq30 neq 0 or
third neq30 neq 0) and
(first neq31 neq 0 or
second neq31 neq 0 or
third neq31 neq 0) and
(first neq32 neq 0 or
second neq32 neq 0 or
third neq32 neq 0) and
(first neq33 neq 0 or
second neq33 neq 0 or
third neq33 neq 0) and
(first neq34 neq 0 or
second neq34 neq 0 or
third neq34 neq 0) and
(first neq35 neq 0 or
second neq35 neq 0 or
third neq35 neq 0) and
(first neq36 neq 0 or
second neq36 neq 0 or
third neq36 neq 0) and
(first neq37 neq 0 or
second neq37 neq 0 or
third neq37 neq 0) and
(first neq38 neq 0 or
second neq38 neq 0 or
third neq38 neq 0) and
(first neq39 neq 0 or
second neq39 neq 0 or
third neq39 neq 0) and
(first neq40 neq 0 or
second neq40 neq 0 or
third neq40 neq 0) and
(first neq41 neq 0 or
second neq41 neq 0 or
third neq41 neq 0) and
(first neq42 neq 0 or
second neq42 neq 0 or
third neq42 neq 0) and
(first neq43 neq 0 or
second neq43 neq 0 or
third neq43 neq 0) and
(first neq44 neq 0 or
second neq44 neq 0 or
third neq44 neq 0) and
(first neq45 neq 0 or
second neq45 neq 0 or
third neq45 neq 0) and
(first neq46 neq 0 or
second neq46 neq 0 or
third neq46 neq 0) and
(first neq47 neq 0 or
second neq47 neq 0 or
third neq47 neq 0) and
(first neq48 neq 0 or
second neq48 neq 0 or
third neq48 neq 0) and
(first neq49 neq 0 or
second neq49 neq 0 or
third neq49 neq 0) and
(first neq50 neq 0 or
second neq50 neq 0 or
third neq50 neq 0) and
(first neq51 neq 0 or
second neq51 neq 0 or
third neq51 neq 0) and
(first neq52 neq 0 or
second neq52 neq 0 or
third neq52 neq 0) and
(first neq53 neq 0 or
second neq53 neq 0 or
third neq53 neq 0) and
(first neq54 neq 0 or
second neq54 neq 0 or
third neq54 neq 0) and
(first neq55 neq 0 or
second neq55 neq 0 or
third neq55 neq 0) and
(first neq56 neq 0 or
second neq56 neq 0 or
third neq56 neq 0) and
(first neq57 neq 0 or
second neq57 neq 0 or
third neq57 neq 0) and
(first neq58 neq 0 or
second neq58 neq 0 or
third neq58 neq 0) and
(first neq59 neq 0 or
second neq59 neq 0 or
third neq59 neq 0) and
(first neq60 neq 0 or
second neq60 neq 0 or
third neq60 neq 0) and
(first neq61 neq 0 or
second neq61 neq 0 or
third neq61 neq 0) and
(first neq62 neq 0 or
second neq62 neq 0 or
third neq62 neq 0) and
(first neq63 neq 0 or
second neq63 neq 0 or
third neq63 neq 0) and
(first neq64 neq 0 or
second neq64 neq 0 or
third neq64 neq 0) and
(first neq65 neq 0 or
second neq65 neq 0 or
third neq65 neq 0) and
(first neq66 neq 0 or
second neq66 neq 0 or
third neq66 neq 0) and
d(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,k(v0,v1))),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(k(k(v2,v1),v2),k(v0,v1))),k(v2,v1))) = 0 and
true;
rlqe
ex(v2c1,
ex(v2c3,
ex(v2c2,phi)));