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};
v3 := {v3c1, v3c2, v3c3};
v3c2 := 0;
neq0 := k(v0,v2);
neq1 := k(v0,k(v3,v2));
neq2 := k(v0,k(v2,v1));
neq3 := k(v0,v3);
neq4 := k(v0,k(k(v2,v1),v2));
neq5 := k(v0,k(v3,k(k(k(k(v2,v1),v2),v0),v0)));
neq6 := k(v1,k(k(k(v2,v1),v2),v0));
neq7 := k(v1,k(k(k(k(v2,v1),v2),v0),v0));
neq8 := k(v1,v2);
neq9 := k(v1,k(v3,v2));
neq10 := k(v1,k(k(v2,v1),v2));
neq11 := k(v1,k(v3,k(k(k(k(v2,v1),v2),v0),v0)));
neq12 := k(k(k(k(v2,v1),v2),v0),v2);
neq13 := k(k(k(k(v2,v1),v2),v0),k(v3,v2));
neq14 := k(k(k(k(v2,v1),v2),v0),k(v2,v1));
neq15 := k(k(k(k(v2,v1),v2),v0),v3);
neq16 := k(k(k(k(v2,v1),v2),v0),k(v3,k(k(k(k(v2,v1),v2),v0),v0)));
neq17 := k(k(k(k(k(v2,v1),v2),v0),v0),v2);
neq18 := k(k(k(k(k(v2,v1),v2),v0),v0),k(v3,v2));
neq19 := k(k(k(k(k(v2,v1),v2),v0),v0),k(v2,v1));
neq20 := k(k(k(k(k(v2,v1),v2),v0),v0),v3);
neq21 := k(k(k(k(k(v2,v1),v2),v0),v0),k(k(v2,v1),v2));
neq22 := k(v2,v3);
neq23 := k(v2,k(v3,k(k(k(k(v2,v1),v2),v0),v0)));
neq24 := k(k(v3,v2),k(v2,v1));
neq25 := k(k(v3,v2),k(k(v2,v1),v2));
neq26 := k(k(v2,v1),v3);
neq27 := k(k(v2,v1),k(v3,k(k(k(k(v2,v1),v2),v0),v0)));
neq28 := k(v3,k(k(v2,v1),v2));
neq29 := k(k(k(v2,v1),v2),k(v3,k(k(k(k(v2,v1),v2),v0),v0)));
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
d(v3,v1) = 0 and
d(k(v3,v2),k(v3,k(k(k(k(v2,v1),v2),v0),v0))) = 0 and
true;
rlqe
ex(v3c1,
ex(v3c3,
ex(v2c1,
ex(v2c3,
ex(v2c2,phi)))));