Wendy Krieger and Richard Klitzing have kindly explained to me how the 72 roots of $E_6$ co-locate with the 240 roots of $E_8$ in 8-space - it's subtle, but really quite simple, as follows:

1. as is well-known, the 240 roots of $E_8$ are instantiated by the 240 vertices of Coxeter's polytope $4$$_2$$_1$ and the 72 roots of $E_6$ are instantiated by the 72 vertices of Coxeter's polytope $1$$_2$$_2$;

2. as is less well-known, the $4$$_2$$_1$ can be decomposed into three objects derived from 8-simplices, where these objects have 84, 72, and 84 vertices each;

3. similarly, the $1$$_2$$_2$ can be decomposed into three objects derived from 5-simplices, where these objects have 21, 30, and 21 vertices each (and where each of the objects with 21 vertices further decomposes into a point plus a 5-simplex)

4. these 21, 30, and 21 co-locate with 21, 30, and 21 of the 84, 72, and 84 respectively.

BUT . . . although Wendy and Richard have thus kindly provided a simple way to think about the "co-location" of $E_8$ and $E_6$ roots in 8-space, this nice visualization does NOT really help us to understand whether points in the $E_8$ lattice do or do not "cluster" around points of the $E_6$ lattice in any interesting way, and if so, how in particular.

So, my personal opinion is STILL that Coxeter's prismatic projection of the usual orthogonal basis in 8-space onto the vectors from the center of an octagonal prism to its vertices (in familiar 3-space) would in fact provide a useful tool for seeing whether or not any interesting clustering does in fact occur.

Wendy Krieger and Richard Klitzing have kindly explained to me how the 72 roots of $E_6$ co-locate with 72 of the 240 roots of $E_8$ in 8-space - it's subtle, but really quite simple, as follows:

1. as is well-known, the 240 roots of $E_8$ are instantiated by the 240 vertices of Coxeter's polytope $4$$_2$$_1$ and the 72 roots of $E_6$ are instantiated by the 72 vertices of Coxeter's polytope $1$$_2$$_2$;

2. as is less well-known, the $4$$_2$$_1$ can be decomposed into three objects derived from 8-simplices, where these objects have 84, 72, and 84 vertices each;

3. similarly, the $1$$_2$$_2$ can be decomposed into three objects derived from 5-simplices, where these objects have 21, 30, and 21 vertices each (and where each of the objects with 21 vertices further decomposes into a point plus a 5-simplex)

4. these 21, 30, and 21 co-locate with 21, 30, and 21 of the 84, 72, and 84 respectively.

BUT . . . although Wendy and Richard have thus kindly provided a simple way to think about the "co-location" of $E_8$ and $E_6$ roots in 8-space, this nice visualization does NOT really help us to understand whether points in the $E_8$ lattice do or do not "cluster" around points of the $E_6$ lattice in any interesting way, and if so, how in particular.

So, my personal opinion is STILL that Coxeter's prismatic projection of the usual orthogonal basis in 8-space onto the vectors from the center of an octagonal prism to its vertices (in familiar 3-space) would in fact provide a useful tool for seeing whether or not any interesting clustering does in fact occur.

Wendy Krieger and Richard Klitzing have kindly explained to me how the 72 roots of $E_6$ co-locate with the 240 roots of $E_8$ in 8-space - it's subtle, but really quite simple, as follows:

1. as is well-known, the 240 roots of $E_8$ are instantiated by the 240 vertices of Coxeter's polytope $4$$_2$$_1$ and the 72 roots of $E_6$ are instantiated by the 72 vertices of Coxeter's polytope $1$$_2$$_2$;

2. as is less well-known, the $4$$_2$$_1$ can be decomposed into three objects derived from 8-simplices, where these objects have 84, 72, and 84 vertices each;

3. similarly, the $1$$_2$$_2$ can be decomposed into three objects derived from 5-simplices, where these objects have 21, 30, and 21 vertices each (and where each of the objects with 21 vertices further decomposes into a point plus a 5-simplex)

4. these 21, 30, and 21 co-locate with 21, 30, and 21 of the 84, 72, and 84 respectively.

BUT . . . although Wendy and Richard have thus kindly provided a simple way to think about the "co-location" of $E_8$ and $E_6$ roots in 8-space, this nice visualization does NOT really help us to understand whether points in the $E_8$ lattice do or not "cluster" around points of the $E_6$ lattice in any interesting way, and if so, how in particular.

So, my personal opinion is STILL that Coxeter's prismatic projection of the usual orthogonal basis in 8-space onto the vectors from the center of an octagonal prism to its vertices (in familiar 3-space) would in fact provide a useful tool for seeing whether or not any interesting clustering does in fact occur.

Wendy Krieger and Richard Klitzing have kindly explained to me how the 72 roots of $E_6$ co-locate with the 240 roots of $E_8$ in 8-space - it's subtle, but really quite simple, as follows:

1. as is well-known, the 240 roots of $E_8$ are instantiated by the 240 vertices of Coxeter's polytope $4$$_2$$_1$ and the 72 roots of $E_6$ are instantiated by the 72 vertices of Coxeter's polytope $1$$_2$$_2$;

2. as is less well-known, the $4$$_2$$_1$ can be decomposed into three objects derived from 8-simplices, where these objects have 84, 72, and 84 vertices each;

3. similarly, the $1$$_2$$_2$ can be decomposed into three objects derived from 5-simplices, where these objects have 21, 30, and 21 vertices each (and where each of the objects with 21 vertices further decomposes into a point plus a 5-simplex)

4. these 21, 30, and 21 co-locate with 21, 30, and 21 of the 84, 72, and 84 respectively.

BUT . . . although Wendy and Richard have thus kindly provided a simple way to think about the "co-location" of $E_8$ and $E_6$ roots in 8-space, this nice visualization does NOT really help us to understand whether points in the $E_8$ lattice do or do not "cluster" around points of the $E_6$ lattice in any interesting way, and if so, how in particular.

So, my personal opinion is STILL that Coxeter's prismatic projection of the usual orthogonal basis in 8-space onto the vectors from the center of an octagonal prism to its vertices (in familiar 3-space) would in fact provide a useful tool for seeing whether or not any interesting clustering does in fact occur.