Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
M.C. Escher, "Cubic Space Division", ©
In contrast to tilings, honeycombs or tesselations of any other dimension, lattices are mere periodic point sets without any further elements and incidence structure. Periodicity here means, that whenever R_{1,0} is some being found point difference vector (R_{1,0} = r_{1}  r_{0}), then not only q = r + R_{1,0} belongs to the same lattice for any lattice point r, but moreover any point q_{k} = r + k R_{1,0} for any k ∈ ℤ. Now consider a corresponding difference vector subset, which makes up a base of the lattice. The whole lattice then happens to be nothing but the ℤspan of that base. The base elements are called the primitive lattice vectors. For conveniance usually a lattice point at the origin is assumed, if not stated explicitely otherwise.
Most often a small change of coordinate values in the primitive vectors does not change the symmetry type of a lattice. A. Bravais not only listed these types for 3D, he also proved that there are exactly 14 of them (in that dimension). Accordingly the representants for either different symmetry type are nowadays called Bravais lattices, even within other dimensions.
A slightly larger vector subset than a mere base, in fact the closure of that base wrt. to reflection, then is called a root system. That is, whenever R and Q belongs to that set, then also does P, the reflection of Q wrt. to a hyperplane perpandicular to R. – Conversely, it is always possible to divide such a root system into 2 complemental subsets, a positive and a negative one. This can be done by selecting from a first direction either R or R and then iteratively continuing this same selectioning under the additional restriction that for R and Q already being selected into one subset, that, whenever R + Q would be a root as well, then R + Q also has to be selected into the same. From these positive roots a still smaller subset of simple roots then can be defined in turn as those ones, which cannot be written as a sum of other positive ones. Such a set of simple roots then represents a lattice base again.
Most authors additionally distinguish between lattices and root lattices. But in view of the former definition of a root system there would be none. In fact those authors would include a further axiom into the notion of a root system. The complemental subset of authors instead would address that very notion as a crystallographic root system. This further axiom requires that for any Q, R from that root system (to be) the value 2 Q·R / R·R has to be an integer. I.e. Q and the reflection of Q wrt. to the hyperplane perpendicular to R have to produce a difference vector, which ought be an integral multiple of R. Accordingly, in view of the former paragraph, any lattice would be a root lattice. Whereas, in view of crystallography, only those, which bow to this additional restriction, can truely be attributed such. – Quite generally, any pair of roots provides that (then not necessarily integral) number. When being restricted to simple roots, i.e. a base, the set of those numbers in effect makes up a matrix, which is called the Cartan matrix. Note that this term, which is used as matrix entry, is clearly not symmetrical in R and Q; therefore the whole Cartan matrix too will not be symmetrical in general. – Sometimes also the mere matrix of the (unweighted) scalarproducts is referenced instead. That then is the associated Gram matrix. That one clearly is always symmetrical.
A further class of lattices is obtained by reciprocation. On the one hand the reciprocal lattice Λ* occurs by means of the Fourier transform of an according wavefunction based on any direct or position lattice Λ as the support for the resulting Bragg peaks in kspace. (By means of the quantum physical p = ℏk, that kspace sometimes also is refered to as momentum space.) In fact, when R is a period of the direct lattice, then also the wave functions of the two position vectors r resp. r + R ought provide the same values. Thus exp(i k·r) = exp(i k·(r+R)) = exp(i k·r) exp(i k·R), the last equal sign here just evaluates the rules of exponentiation, and therefore exp(i k·R) = 1. – Alternatively, and without that crystallographic referenece to Fourier transforms of wavefunctions, the reciprocal lattice can likewise be defined directly as Λ* = {k  k·R∈ℤ for all R∈Λ} (which then happens to coincide with the above description up to a global scaling factor of 2π). Within dimensions greater than 1 this k·R happens to be a scalar product, showing that generally corresponding base vectors of either space each have to have the same directions, but mutually being of inverse absolute size. – On the other hand this very mathematical description shows that the reciprocal of the reciprocal lattice then again happens to be the direct lattice again. And so there is no intrinsic preference between either one.
Wrt. (crystallographic) root lattices Λ the reciprocal lattice Λ* also is called a weight lattice. Right in this context it happens, that the root lattice itself always is a sublattice of its weight lattice. Moreover, the corresponding index  Λ* / Λ  is readily accessible by the determinant of the Cartan matrix of Λ. Weight lattices need no longer be crystallographic too, their Cartan matrix entries in general will become rationals. The exceptional lattices, where both are crystallographic, i.e. where  Λ* / Λ  = 1 (like for E_{8}), are called unimodular.
Despite the fact that a lattice is a mere point set, there is a way how to associate a corresponding tesselation too. In fact, the Voronoi complex V(Λ) is a tesselation, defined by the individual Voronoi cells V_{R}(Λ) are the closed subsets of embedding space (ℝspan of Λ), which has lower or equal distance to some given lattice point R, than to any other one. The vertex set of the Voronoi complex, also refered to as the set of lattice holes, needs not be a lattice. That Voronoi complex moreover can be dualised (within that embedding space), resulting in the associated Delone complex D(Λ). The vertex set of this Delone complex for sure is the starting lattice again.
While within direct spaces to crystallographers the individual Voronoi cell also is known as WignerSize cell of the respective lattice, in reciprocal spaces it then is called its (first) Brillouin zone.
Root lattices are closely related to Weyl groups. In fact, the rotational and translatoric symmetries of a lattice are described by the associated Weyl group. Conversely, just as the finite Coxeter groups provide polytopes, the infinite Weyl groups also provide flat ndimensional euclidean tesselations, some of which reoccur here as Delone complexes of a corresponding lattice. It is a matter of fact, that Weyl groups and the associated lattices generally are labeled exactly the same by the first few capital letters of the alphabet and the dimension of the spanned space is given as its subscript. (Some authors however try to distinguish this a little by using different fonts for either usage.)
It shall be warned here though, that the usage of B_{n} and C_{n} in literature tends to be mutually inverted sometimes, both from author to author, or even within a single monograph from chapter to chapter. The here provided attribution of Weyl groups (and hence for the to be assciated lattices) however is in accordance to Coxeter.
Lattice  Simple Roots & Notes  Count & Hull of Roots  Cartan matrix 
Weyl group  Voronoi complex V / Voronoi cell V_{0}  Delone complex D / Delone cells D_{k}  
A_{n} (n ≥ 1) 
The simplest way to
describe A_{n} makes use of
A_{1} ≅ I_{1} ≅ ℤ  A_{n}* / A_{n}  = n+1 
1D: 2  {} 2D: 6  {6} 3D: 12  co 4D: 20  spid 5D: 30  scad 6D: 42  staf 7D: 56  suph 8D: 72  soxeb ... nD: n(n+1)  x3o3o...3o3x (n nodes, n≥1) expanded simplex 
2 1 0 0 0 ... 1 2 1 0 0 ... 0 1 2 1 0 ... 0 0 1 2 1 ... 0 0 0 1 2 ... ... 
o3o3o...3o3*a (n+1 nodes) 
V(A_{1}) ≅ aze V_{0}(A_{1}) ≅ {} V(A_{2}) ≅ hexat V_{0}(A_{2}) ≅ {6} V(A_{3}) ≅ radh V_{0}(A_{3}) ≅ rad ..._{ } ≅ ... V_{0}(A_{4}) ≅ duspid V_{0}(A_{n}) ≅ xo..oo3ox..oo3...3oo..xo3oo..ox&#zy _{ } (n node positions, n layers) _{ } where y = x/sqrt(n+1) _{ } : projection of (n+1)dim. _{ } hypercube along bodydiagonal _{ } : m3o...o3m (i.e. dual of x3o...o3x) 
D(A_{1}) ≅ aze D_{1}(A_{1}) ≅ {} D(A_{2}) ≅ trat D_{1}(A_{2}) ≅ {3} _{ } D_{2}(A_{2}) ≅ dual {3} D(A_{3}) ≅ octet D_{1}(A_{3}) ≅ tet _{ } D_{2}(A_{3}) ≅ oct _{ } D_{3}(A_{3}) ≅ dual tet D(A_{4}) ≅ cypit D_{1}(A_{4}) ≅ pen _{ } D_{2}(A_{4}) ≅ rap _{ } D_{3}(A_{4}) ≅ inv rap _{ } D_{4}(A_{4}) ≅ dual pen D(A_{5}) ≅ cyxh D_{1}(A_{5}) ≅ hix _{ } D_{2}(A_{5}) ≅ rix _{ } D_{3}(A_{5}) ≅ dot _{ } D_{4}(A_{5}) ≅ inv rix _{ } D_{5}(A_{5}) ≅ dual hix D(A_{6}) ≅ cyloh D_{1}(A_{6}) ≅ hop _{ } D_{2}(A_{6}) ≅ ril _{ } D_{3}(A_{6}) ≅ bril _{ } D_{4}(A_{6}) ≅ inv bril _{ } D_{5}(A_{6}) ≅ inv ril _{ } D_{6}(A_{6}) ≅ dual hop ... D(A_{n}) ≅ x3o3o...3o3*a _{ } (n+1 nodes, n>1) D_{k}(A_{n}) : kth layer section of (n+1)dim. _{ } _{ } hypercube perp. to bodydiagonal  
C_{n} (n ≥ 2) 
C_{n} ≅ (A_{1})^{n} ≅ ℤ^{n}
C_{2} ≅ ℤ[i] = {a+bi  a,b∈ℤ}  C_{n}* / C_{n}  = 2 (C_{1} = I_{1} ≅ A_{1} ≅ ℤ) 
large roots: 2D: 4  {4} 3D: 12  co 4D: 24  ico 5D: 40  rat 6D: 60  rag 7D: 84  rez 8D: 112  rek 9D: 144  riv 10D: 180  rake ... nD: 2n(n1)  o3x3o...3o4o (n nodes, n>2) rectified crosspolytope small roots just point to: 2D: 4  side midpoints of {4} 3D: 6  centers of {4} of co 4D: 8  centers of some oct of ico 5D: 10  centers of hex of rat 6D: 12  centers of tac of rag 7D: 14  centers of gee of rez 8D: 16  centers of zee of rek 9D: 18  centers of ek of riv 10D: 18  centers of vee of rake ... nD: 2n  centers of . x3o...3o4o of o3x3o...3o4o inscribed crosspolytope 
... ... 2 1 0 0 0 ... 1 2 1 0 0 ... 0 1 2 1 0 ... 0 0 1 2 2 ... 0 0 0 1 2 
o4o3o3o...3o4o (n+1 nodes, n>1) 
V(C_{2}) ≅ squat V_{0}(C_{2}) ≅ {4} V(C_{3}) ≅ chon V_{0}(C_{3}) ≅ cube V(C_{4}) ≅ test V_{0}(C_{4}) ≅ tes V(C_{5}) ≅ penth V_{0}(C_{5}) ≅ pent V(C_{6}) ≅ axh V_{0}(C_{6}) ≅ ax ... V(C_{n}) ≅ x4o3o3o...3o4o _{ } (n+1 nodes, n>1) _{ } : hypercubical honeycomb 
D(C_{2}) ≅ squat D_{1}(C_{2}) ≅ {4} D(C_{3}) ≅ chon D_{1}(C_{3}) ≅ cube D(C_{4}) ≅ test D_{1}(C_{4}) ≅ tes D(C_{5}) ≅ penth D_{1}(C_{5}) ≅ pent D(C_{6}) ≅ axh D_{1}(C_{6}) ≅ ax ... D(C_{n}) ≅ x4o3o3o...3o4o _{ } (n+1 nodes, n>1) _{ } : hypercubical honeycomb  
B_{n} (n ≥ 3) 
B_{n} ≅ {∑c_{i}e_{i} ∈ ℤ^{n+1}  ∑c_{i} even}
B_{3} ≅ A_{3} ≅ fcc  B_{n}* / B_{n}  = 2 (B_{2} = A_{1}×A_{1} ≅ C_{2}) © 
large roots: 3D: 6  oct 4D: 8  hex 5D: 10  tac 6D: 12  gee 7D: 14  zee 8D: 16  ek 9D: 18  vee 10D: 20  ka ... nD: 2n  x3o3o...3o4o (n nodes, n>1) crosspolytope small roots just point to: 3D: 12  edge midpoints of oct 4D: 24  edge midpoints of hex 5D: 40  edge midpoints of tac 6D: 60  edge midpoints of gee 7D: 84  edge midpoints of zee 8D: 112  edge midpoints of ek 9D: 144  edge midpoints of vee 10D: 180  edge midpoints of ka ... nD: 2n(n1)  centers of x . .... . . of x3o3...o3o4o inscribed rectified crosspolytope 
... ... 2 1 0 0 0 ... 1 2 1 0 0 ... 0 1 2 1 0 ... 0 0 1 2 1 ... 0 0 0 2 2 
o3o3o *b3o...3o4o (n+1 nodes, n>3) 
V(B_{3}) ≅ radh V_{0}(B_{3}) ≅ rad V(B_{4}) ≅ icot V_{0}(B_{4}) ≅ ico ... V_{0}(B_{n}) ≅ qo3oo3...3oo4ox&#zy _{ } _{ } (n node positions) _{ } _{ } where y = x sqrt(n+1)/2 
D(B_{3}) ≅ octet D_{1}(B_{3}) ≅ tet _{ } D_{2}(B_{3}) ≅ oct D(B_{4}) ≅ hext D_{1}(B_{4}) ≅ hex D(B_{5}) ≅ hinoh D_{1}(B_{5}) ≅ hin _{ } D_{2}(B_{5}) ≅ tac D(B_{6}) ≅ haxh D_{1}(B_{6}) ≅ hax _{ } D_{2}(B_{6}) ≅ gee ... D(B_{n}) ≅ x3o3o *b3o...3o4o _{ } (n+1 nodes, n>2) _{ } : demihypercubical honeycomb  
D_{n} (n ≥ 4) 
D_{n} ≅ {∑c_{i}e_{i} ∈ ℤ^{n}  ∑c_{i} even} D_{4} ≅ F_{4}  D_{n}* / D_{n}  = 4 (D_{3} = A_{3} ≅ B_{3} ≅ fcc) 
4D: 24  ico 5D: 40  rat 6D: 60  rag 7D: 84  rez 8D: 112  rek 9D: 144  riv 10D: 180  rake ... nD: 2n(n1)  o3x3o...3o4o (n nodes, n>2) rectified crosspolytope 
... ... 2 1 0 0 0 ... 1 2 1 0 0 ... 0 1 2 1 1 ... 0 0 1 2 0 ... 0 0 1 0 2 
o3o3o o3o3o *b3o...3*e (n+1 nodes, n>4) resp. o3o3o *b3o *b3o (n=4) 
V(D_{4}) ≅ icot V_{0}(D_{4}) ≅ ico ... V_{0}(D_{n}) ≅ xoo3ooo3oxo *b3ooo...ooo3oox&#zy _{ } _{ } (n node positions) _{ } _{ } where y = x sqrt[(n+1)/8] 
D(D_{4}) ≅ hext D_{1}(D_{4}) ≅ hex D(D_{5}) ≅ hinoh D_{1}(D_{5}) ≅ hin _{ } D_{2}(D_{5}) ≅ tac D(D_{6}) ≅ haxh D_{1}(D_{6}) ≅ hax _{ } D_{2}(D_{6}) ≅ gee ... D(D_{n}) ≅ x3o3o o3o3o *b3o...3*e _{ } (n+1 nodes, n>4) _{ } : demihypercubical honeycomb  
E_{6} 
 E_{6}* / E_{6}  = 3
(E_{5} = D_{5} 
72  mo  o3o3o3o3o *c3x Gosset polypeton 1_{2,2} 
2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 1 0 0 0 1 2 0 0 0 0 1 0 2 1 0 0 0 0 1 2 
o3o3o3o3o *c3o3o 
V(E_{6}) ≅ reciprocal(2_{2,2}) 
D(E_{6}) ≅ jakoh D_{1}(E_{6}) ≅ jak Gosset hexacomb 2_{2,2} Gosset polypeton 2_{2,1} _{ } D_{2}(E_{6}) ≅ gyrated jak _{ } Gosset polypeton 2_{1,2}  
E_{7} 
 E_{7}* / E_{7}  = 2 
126  laq  x3o3o3o *c3o3o3o Gosset polyexon 2_{3,1} 
2 1 0 0 0 0 0 1 2 1 0 0 0 0 0 1 2 1 0 0 0 0 0 1 2 1 1 0 0 0 0 1 2 0 0 0 0 0 1 0 2 1 0 0 0 0 0 1 2 
o3o3o3o3o3o3o *d3o 
... 
D(E_{7}) ≅ naquoh D_{1}(E_{7}) ≅ oca Gosset heptacomb 3_{3,1} Gosset polyexon 3_{3,0} _{ } (= simplex) _{ } D_{2}(E_{7}) ≅ naq _{ } Gosset polyexon 3_{2,1}  
E_{8} 
 E_{8}* / E_{8}  = 1 
240  fy  o3o3o3o *c3o3o3o3x Gosset polyzetton 4_{2,1} 
2 1 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 1 2 1 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 0 2 1 0 0 0 0 0 0 1 2 
o3o3o3o *c3o3o3o3o3o 
... 
D(E_{8}) ≅ goh D_{1}(E_{8}) ≅ ene Gosset octacomb 5_{2,1} Gosset polyzetton 5_{2,0} _{ } (= simplex) _{ } D_{2}(E_{8}) ≅ ek _{ } Gosset polyzetton 5_{1,1} _{ } (= crosspolytope)  
F_{4} 
F_{4} ≅ B_{4} ≅ D_{4}  F_{4}* / F_{4}  = 1 
large roots: 24  ico small roots: 24  centers of oct of ico inscribed dually oriented ico 
2 1 0 0 1 2 2 0 0 1 2 1 0 0 1 2 
o3o3o4o3o 
V(F_{4}) ≅ icot _{ } : icositetrachoric tetracomb 
D(F_{4}) ≅ hext _{ } : hexadecachoric tetracomb  
G_{2} 
One way to
describe G_{2} makes use of
G_{2} ≅ A_{2}  G_{2}* / G_{2}  = 1 
large roots: 6  u6o small roots: 6  o6x dual hexagon of halved sides 
2 1 3 2 
o3o6o 
V(G_{2}) ≅ hexat = o3o6x 
D(G_{2}) ≅ trat = x3o6o  
I_{1} 
I_{1} = ℤ ≅ A_{1} ≅ B_{1} ≅ C_{1}  I_{1}* / I_{1}  = 1 
2  x 
2 
o∞o 
V(I_{1}) ≅ o∞x 
D(I_{1}) ≅ x∞o  
reducible Λ × Λ' 

N(Λ)+N(Λ')  ... tegum product of components 
Cart(Λ) 0 0 Cart(Λ') 
Weyl(Λ) x Weyl(Λ') 
V(Λ x Λ') ≅ V(Λ) x V(Λ') honeycomb product of components 
D(Λ x Λ') ≅ D(Λ) x D(Λ') honeycomb product of components 
The latter entry shows that the individual Voronoi cells will be isohedral, iff the lattice is a root lattice or a direct product of identical root lattices.
A directly constructed base for the dual lattices can easily be derived from its associated original one: If B is the matrix of column vectors of the original lattice base, then A = (B^{1})^{T} clearly is the matrix of column vectors of the reciprocal lattice. (For, the scalar products, required in the definition of the reciprocal lattice, are just the entries of the matrix A^{T}B, which, by the above choice, just happens to be the identity matrix.)
On the other hand, for weight lattices we know already that the corresponding root lattice defines a subset of points. Thus it might be more direct to provide the further generators only, instead.
Lattice  Quotient  Further generator(s)  Overlay as compounded vertex set 
Voronoi complex V / Voronoi cell V_{0}  Delone complex D / Delone cells D_{k}  
A_{n}* (n ≥ 1) 
A_{n}* / A_{n} ≅ ℤ_{n+1}
Just as A_{n} was best
A_{1}* ≅ A_{1} ≅ ℤ 

xoo...o3oxo...o3oox...o ... ooo...x3*a (n+1 compound components, n+1 node positions) 
V(A_{1}*) ≅ aze V_{0}(A_{1}*) ≅ {} V(A_{2}*) ≅ hexat V_{0}(A_{2}*) ≅ {6} V(A_{3}*) ≅ batch V_{0}(A_{3}*) ≅ toe V(A_{4}*) ≅ otcypit V_{0}(A_{4}*) ≅ gippid V(A_{5}*) ≅ gapcyxh V_{0}(A_{5}*) ≅ gocad V(A_{6}*) ≅ gapcyloh V_{0}(A_{6}*) ≅ gotaf ... V(A_{n}*) ≅ x3x3...x3*a (n+1 nodes) 
D(A_{1}*) ≅ aze D(A_{2}*) ≅ trat ...  
C_{n}* (n ≥ 2) 
C_{n}* / C_{n} ≅ ℤ_{2} bodycentered hypercubic
C_{2}* ≅ C_{2} (rotoscaled) 

xo4oo3oo ... oo4ox 
V(C_{2}*) ≅ squat V_{0}(C_{2}*) ≅ {4} V(C_{3}*) ≅ batch V_{0}(C_{3}*) ≅ toe V(C_{4}*) ≅ icot V_{0}(C_{4}*) ≅ ico V(C_{5}*) ≅ titanoh V_{0}(C_{5}*) ≅ bittit V(C_{6}*) ≅ traxh V_{0}(C_{6}*) ≅ brag ... V(C_{2m}*) ≅ o4o3o...o3x3o...o3o4o V(C_{2m+1}*) ≅ o4o3o...o3x3x3o...o3o4o(m>1 nodes unringed on either side) V(C_{2}*) ≅ o4x4o V(C_{3}*) ≅ o4x3x4o(m=1) 
D(C_{2}*) ≅ squat ...  
B_{n}* (n ≥ 3) 
B_{n}* / B_{n} ≅ ℤ_{2} primitive hypercubic B_{n}* ≅ C_{n} ≅ (A_{1})^{n} ≅ ℤ^{n} 

xo3oo3ox *b3oo...oo4oo(n+1 node positions, n≥3) 
V(B_{3}*) ≅ chon V_{0}(B_{3}*) ≅ cube V(B_{4}*) ≅ test V_{0}(B_{4}*) ≅ tes V(B_{5}*) ≅ penth V_{0}(B_{5}*) ≅ pent V(B_{6}*) ≅ axh V_{0}(B_{6}*) ≅ ax ... V(B_{n}*) ≅ o3o3o *b3o...o3o4x 
D(B_{3}*) ≅ chon D_{1}(B_{3}*) ≅ cube D(B_{4}*) ≅ test D_{1}(B_{4}*) ≅ tes D(B_{5}*) ≅ penth D_{1}(B_{5}*) ≅ pent D(B_{6}*) ≅ axh D_{1}(B_{6}*) ≅ ax ...  
D_{n}* (n ≥ 4) 
D_{n}* / D_{n} ≅ ℤ_{4} (n odd) bodycentered hypercubic D_{n}* ≅ C_{n}* 

xooo3oooo3oxoo ooxo3oooo3ooox *b3oooo....oooo3*e(n+1 node positions, n>4) xooo3oooo3oxoo *b3ooxo *b3ooox(n=4) 
V(D_{4}*) ≅ icot V_{0}(D_{4}*) ≅ ico V(D_{5}*) ≅ titanoh V_{0}(D_{5}*) ≅ bittit V(D_{6}*) ≅ traxh V_{0}(D_{6}*) ≅ brag ... V(D_{2m}*) ≅ o3o3o o3o3o *b3o...o3x3o...o3*e V(D_{2m+1}*) ≅ o3o3o o3o3o *b3o...o3x3x3o...o3*e(m>2 nodes unringed on either side) V(D_{4}*) ≅ o3x3o *b3o *b3o V(D_{5}*) ≅ o3x3o o3x3o *b3*e(m=2) 
...  
E_{6}* 
E_{6}* / E_{6} ≅ ℤ_{3} 

xoo3ooo3ooo3ooo3oxo *c3ooo3oox 
V(E_{6}*) ≅ ramoh V_{0}(E_{6}*) ≅ ram Gosset hexacomb 0_{2,2,2} Gosset polypeton 0_{2,2,1} 
...  
E_{7}* 
E_{7}* / E_{7} ≅ ℤ_{2} 

xo3oo3oo3oo3oo3oo3ox *d3oo 
V(E_{7}*) ≅ linoh V_{0}(E_{7}*) ≅ lin Gosset heptacomb 1_{3,3} Gosset polyexon 1_{3,2} 
... 
The cases E_{8}* ≅ E_{8}, F_{4}* ≅ F_{4}, G_{2}* ≅ G_{2}, and I_{1}* ≅ I_{1} can be omitted in the above table for obvious reasons.
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