Geodesic
Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 5-68

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The present paper is devoted to a detailed computer study of the action of Chevalley group $G(\mathrm E_6,R)$ on the minimal module $V(\varpi_1)$. Our main objectives are an explicit choice and tabulation of the signs of structure constants for this action, compatible with the choice of positive Chevalley base, construction of multilinear invariants and equations on the matrix entries of matrices from $G(\mathrm E_6,R)$ in this representation, and explicit tabulation of root elements. 
@article{ZNSL_2006_338_a0,
     author = {N. A. Vavilov and A. Yu. Luzgarev and I. M. Pevzner},
     title = {Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--68},
     publisher = {mathdoc},
     volume = {338},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_338_a0/}
}
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N. A. Vavilov; A. Yu. Luzgarev; I. M. Pevzner. Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 5-68. http://geodesic.mathdoc.fr/item/ZNSL_2006_338_a0/