Mapping the geometry of the E6 group Page: 4 of 31
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3 The construction of the group E6
3.1 The generalized Euler parametrization for E6
To give an Euler parametrization for E6 we start by choosing its maximal subgroup. It is H F4,
the group generated by the first 52 matrices. Let 'P be the linear complement of f4 in e6. We then
search for a minimal linear subset V of ', which, under action of Ad(F4), generates the whole '.
Looking at the structure constants we see that V can be chosen as the linear space generated by
c53, c70. Note that they commute.
If we then write the general element g of E6 in the form
g exp(h) exp(v) exp(h) , h, h E f4 , v E V , (3.1)
we have a redundancy of dimension 28. As argued in  we expect to find a 28-dimensional
subgroup of F4 ,which, acting by adjunction, defines an automorphism of V. This can be done by
noticing that V commutes with the first 28 matrices c,, i 1,..., 28, which generate an SO(8)
subgroup of F4.
We found convenient to introduce the change of base
:53 + , (3.2)
C70 2 c53 + 2c7 . (3.3)
Thus we have
g[xi, . . . , X78] B[xi, . .. ,X24]cX25C3X26C70 F4[X27, . . .78]
where B F4/SO(8). We chose for F4 the Euler parametrization given in  so that we found for
B[x1, . . .X24] = BF4 [1X, - - , x16]B9 [17, . - , , 23 (3.4)
BF4[x1, [xl , x15 9 [x1, . . . ,x7 E BSC4s8 [X9, . . . ,1 o 6c22 , (3.5)
B9x .. 6 1e2C1e6,X2C15 6X4C25 6X5C5 6X6C1 6X7C20(36
B [x1, .. ., 7] --1a*26r~s*23es5rdes (3.6)
B8[x1, ... 2] e e 62C165eX3 6C2535e5 6EX6C1 (3.7)
and the tilded matrices are the ones introduced in .
3.2 Determination of the range for the parameters.
To determine the range of the parameters we will use the topological method we have developed
in . Let us first determine the volume of E6 by means of the Macdonald formula.
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Cerchiai , Bianca; Bernardoni, Fabio; Cacciatori, Sergio L.; Cerchiai, Bianca L. & Scotti, Antonio. Mapping the geometry of the E6 group, article, October 1, 2007; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc895124/m1/4/: accessed October 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.