The Clifford Algebra Cl(8) has dimension 2^8 = 256.

Since 256 = 16 x 16 = 2^4 x 2^4, the full spinors of Cl(8) are16-dimensional, and **the half-spinors of Cl(8) are8-dimensional**.

Cl(8) has graded structure

1 8 28 56 70 56 28 8 1

**The 28-dimensional grade-2 bivectors of Cl(8) form the LieAlgebra Spin(8)**.

The **E8**** Lie Algebra** isthe sum (on this page I am using the word "sum" very imprecisely)of

and

one 2^7 = **128-dimensional Cl(16) half-spinor space**.

**First**, construct the 120-dimensional Spin(16) fromCl(8):

Since Cl(16) can be written as the tensor product Cl(16) = Cl(8) xCl(8), the graded structure of Cl(16) can be written in terms of thegraded structure of Cl(8) as follows:

1 8 28 56 70 56 28 8 1 8 64 224 448 560 448 224 64 8 28 224 784 1568 1960 1568 784 224 28 56 448 1568 3136 3920 3136 1568 448 56 70 560 1960 3920 4900 3920 1960 560 70 56 448 1568 3136 3920 3136 1568 448 56 28 224 784 1568 1960 1568 784 224 28 8 64 224 448 560 448 224 64 8 1 8 28 56 70 56 28 8 11 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

Therefore:

Spin(16) = 120 = 28x1 + 8x8 + 1x28 = 28 + 64 + 28, and

**plus the square of the 8-dimensional Cl(8) grade-1 vectorspace**.

**Second**, construct the 128-dimensional half-spinors ofCl(16) from Cl(8):

The Cl(8) half-spinors are 8-dimensional, so that the Cl(8)full-spinor space is 8e + 8o = 16-dimensional, where 8e is onehalf-spinor 8-dimensional space and 8o is the other mirror imagehalf-spinor 8-dimensional space.

ou can construct the spinor space of Cl(16) as the tensor product( 8e + 8o ) x ( 8e + 8o ) as follows:

8ex8e + 8ex8o 8ox8e + 8ox8o

If ee and oo correspond to e, and if eo and oe correspond to o, wehave:

64e + 128o + 64e

so that the Cl(16) spinors are (64 + 64)e + 128o =256-dimensional, and

Therefore:

and

The second 28-dimensional Spin(8) corresponds to the OctonionicSpin(8) symmetries of each of the two half-spinor spaces and of thevector space. Note that the same Octonionic Spin(8) is used for eachof the two half-spinor spaces and for the vector space, as isconsistent with the fact (noted by GeoffreyDixon in his book on Divison Algebras) that the left-adjoint andright-adjoint actions of the Octonions are both isomorphic to eachother, and are also isomorphic to the 8x8 matrix algebra over a real8-dim vector space.

E7 is the sum of 28-dim Spin(8) plus Quaternionic versions of thetwo 8-dim half-spinor spaces of Cl(8) and the 8-dim vector space ofCl(8), each of which is 4x8 = 32-dimensional.

The central 9 elements correspond to the Quaternionic SU(2)symmetries of each of the two half-spinor spaces and of the vectorspace. Note that distinct Quaternionic SU(2)s are used for each ofthe half-spinor spaces, as is consistent with the fact (noted byGeoffrey Dixon in his book on DivisonAlgebras) that the left-adjoint and right-adjoint actions of theQuaternions are not isomorphic to each other, nor are they 4x4 matrixalgebras over a real 4-dim vector space.

E6 is the sum of 28-dim Spin(8) plus Complex versions of the two8-dim half-spinor spaces of Cl(8) and the 8-dim vector space ofCl(8), each of which is 2x8 = 16-dimensional.

The central 2 elements correspond to the Complex U(1) symmetriesof the spinor spaces and of the vector space. Note that the sameComplex U(1) is used for both of the half-spinor spaces, as isconsistent with the fact (noted by GeoffreyDixon in his book on Divison Algebras) that the left-adjoint andright-adjoint actions of the Complex numbers are both isomorphic toeach other, but are not 2x2 matrix algebras over a real 2-dim vectorspace.

D5 is the sum of 28-dim Spin(8) plus a Complex versions of the8-dim vector space of Cl(8), which is 2x8 = 16-dimensional. D5 is theSpin(10) Lie algebra of the Clifford Algebra Cl(10), which factors bytensor product into Cl(10) = Cl(2) x Cl(8). Since the gradedstructure of Cl(2) is 1 + 2 + 1, the graded structure of Cl(10)is

1 8 28 56 70 56 28 8 1 2 16 56 112 140 112 56 16 2 1 8 28 56 70 56 28 8 11 10 45 120 210 252 210 120 45 10 1

The 1 of the 45 corresponds to the grade-2 bivector of Cl(2),acting as U(1) on the 2-dim vector space of Cl(2). It is representedon the pattern by the central element corresponding to the ComplexU(1) symmetry of the Complex version of the Cl(8) vector space.

D4 is the 28-dim Spin(8) of the Cl(8) Clifford Algebra with gradedstructure

1 8 28 56 70 56 28 8 1

B4 is the sum of 28-dim Spin(8) plus the 8-dim vector space ofCl(8). B4 is the Spin(9) Lie algebra of the Clifford Algebra Cl(9),which factors by tensor product into Cl(9) = Cl(1) x Cl(8). Since thegraded structure of Cl(1) is 1 + 1, the graded structure of Cl(9)is

1 8 28 56 70 56 28 8 1 1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1

F4 is the sum of 28-dim Spin(8) plus the two 8-dim half-spinorspaces of Cl(8) and the 8-dim vector space of Cl(8).

Note that the 16-dimensional symmetric space F4 / Spin(9) is theOctonion Projective Plane, which isrepresented on the pattern by the two 8-dim half-spinor spaces ofCl(8).

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