The second Seeley–DeWitt coefficient of the scalar Laplacian on the contact orbit E₈/(E₇·U(1))

We compute the second Seeley–DeWitt heat-kernel coefficient a₂ of the scalar Laplacian on the 114-dimensional contact orbit 𝒪 = E₈/(E₇·U(1)), the adjoint variety of E₈ and the twistor space of the Wolf space EIX = E₈/(E₇ × Sp(1)). Unlike its base, 𝒪 is not symmetric and carries a one-parameter family g_x of invariant metrics, so the computation is organised around the full Nomizu connection. The key ingredient is that the two-dimensional isotropy summand consists of two of the three generators of the Sp(1) factor of the isotropy algebra of EIX: the induced endomorphisms of the 112-dimensional summand obey a Clifford relation, every curvature contraction becomes a trace of a word in them, and the only non-trivial such trace vanishes. We obtain the Ricci curvature and the two invariant Einstein metrics (x = 2, Kähler, and x = 2/29), the closed form ‖Riem‖²(x) = 602x² − 2436x + 21672 + 4x⁻², and a₂(𝒪, g_x) = (1/360)(1974x² − 238392x + 14104608 + 33600x⁻¹ + 24x⁻²), whose x-independent part equals 360 a₂(EIX) identically. For the metric induced from the embedding 𝒪 ⊂ e₈, which is not Einstein, a₂ = 2931339/80 ≈ 36641.74.