Lemma: Change of a self-dualizing pairing

Suppose two positive-definite inner products $h_1,h_2$ make the same cone $K$ self-dual. There is an $h_1$-self-adjoint, positive-definite operator $A\in \mathrm{Aut}(K)$ such that

Proof. Finite-dimensional linear algebra gives the stated positive operator $A$. Its self-adjointness implies

Both dual cones equal $K$, so $AK=K$.