Noncrystallographic Coxeter group H₄ in E₈

The E₈ lattice is constructed in terms of icosians by matching two sets of F₄ lattices described by quaternions. Embedding the noncrystallographic group H₄ into the Weyl group W(E₈) has been described using matrix generators with an emphasis on the relevant Coxeter elements. The conjugacy classes of H₄ in terms of quaternions and the characters of the two four-dimensional irreducible representations are explicitly calculated.