Florin,
Right now I worry less about the nonassociative aspects of E_8 and am more focused on the automorphism and centralizer groups G_2 and F_4.
Nonassociativity is a bit strange. Yet all it says is that (e_ie_j)e_k - e_i(e_je_k) = C_{ijk}^le_l, where the last term is by multiplication table rules e_i(e_je_k). So you think of this as a sort of π/2 phase shift. The physical meaning I think involves the S-matrix. The S-matrix acts on a set of vertices or particles p_i
|φ) = |p_1, p_2, ..., p_i, ..., p_j, ...p_n)
and converts this channel into an S-channel which has some overlap with
(φ| = (p_1, p_2, ..., p_j, ..., p_i, ...p_n|,
so the expectation of the S matrix for these two ordered sets of states is
( S ) = |φ> = (p_1, p_2, ..., p_j, ..., p_j, ...p_n|S|p_1, p_2, ..., p_i, ..., p_j, ...p_n).
By S = 1 2πT this is determined by a transition matrix, which by the exchange of vertices determines the S-T-U relationships or Mandelstam variables. In this case we simply have an exchange or a commutator in a quantum group. This might be represented by (ab)---(ba) as a braid link, and for multiple exchanges the S-matrix determines a braid group. For the exchange of three elements this gives a Yang-Baxter equation which is equivalent to a Jacobi identity on the double commutator [a, [b, c]] and it is equal to zero. The channel is produced by a product of Hilbert spaces for each vertex, so
|p_1, p_2, p_3> = |p_1>|p_2>|p_3>.
Nonassociativity is an ambiguity which says that
|(p_1, p_2), p_3> = (|p_1>|p_2>)|p_3> = |p_1>(|p_2>|p_3>) |C_{123}^4p_4>.
This is a type of Hopf algebraic system, but with a "twist." The noncommutative system is defined by the K-linear map on the vector space V, or between V and V', a multiplication and co-multiplication rule you get the Hopf hexagon. However, for three elements and an associative rule there is a corresponding pentagon (Stasheff polygon), the hexagon and pentagon are fused together to form a general polytope. I can delve into these detail later if you are so interested.
What would this correspond to physically? The ordering ambiguity means there are two S-matrix channels which are not commensurate with each other. For the case of a black hole a string, which is really an S-matrix element, is observed to exhibit completely different physics according to an observer who witnesses it fall towards the black hole from a distance, what Susskind calls a fiducial observer or FIFO, and an observer who falls in with the string, a freely falling observer FREFO. What Susskind has argued is how the physics observed by the two is completely different, but both are physically valid. The general polytope I mention indicates how noncommutative geometry and nonassociative algebra are related (a bit complicated), and physically this means what we think of as a proper ordering of events or trajectories must be "liberalized."
Cheers LC