Consider the root system of a Kac-Moody algebra. Designate with $ alpha_i $ the simple root associated with the node $ i $

from for $ i in {1, ldots, n-1 } $ and from $ beta $ the simple root associated with $ n $,

The dynkin diagram for $ widetilde {E} _8 $ is

begin {align}

circ – circ – & circ – circ – circ – circ – circ – circ \

& | \

& bullet

end

from where $ bullet $ corresponds to the simple root $ beta $, The degree of a root is the coefficient of the root at $ beta $,

Is there evidence of imaginary type roots? $ widetilde {E} _8 $? I only find an imaginary root $ gamma = 3 beta + 2 alpha_1 + 4 alpha_2 + 6 alpha_3 + 5 alpha_4 + 4 alpha_5 + 3 alpha_6 + 2 alpha_7 + alpha_8 $, The origin $ gamma $ has grad $ 3 $,

Many thanks.