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The exchange-only virial relation due to Levy and Perdew is revisited.Invoking the adiabatic connection, we introduce the exchange energy in terms ofthe right-derivative of the universal density functional w.r.t. the couplingstrength $\lambda$ at $\lambda=0$. This agrees with the Levy-Perdew definitionof the exchange energy as a high-density limit of the full exchange-correlationenergy. By relying on $v$-representability for a fixed density at varyingcoupling strength, we prove an exchange-only virial relation without anexplicit local-exchange potential. Instead, the relation is in terms of a limit($\lambda \to 0+$) involving the exchange-correlation potential$v_\mathrm{xc}^\lambda$, which exists by assumption of $v$-representability. Onthe other hand, a local-exchange potential $v_\mathrm{x}$ is not warranted toexist as such a limit.

Exchange-only virial relation from the adiabatic connection