Diagonalizability of adjoint-abelian operators in smooth Minkowski spaces

Let a smooth Minkowski space be a finite-dimensional real normed space whose norm is smooth away from the origin, and let an adjoint-abelian operator be a linear operator $A$ satisfying

$$[Ax,y]=[x,Ay]$$

for all vectors $x,y$, where $[\cdot,\cdot]$ denotes the semi-inner product associated with the norm. Diagonalizability conjecture. Every adjoint-abelian operator of a smooth Minkowski space is diagonalizable. The source presents this as a consequence of a cited theorem, so the claim may already be resolved; the supplied parser status does not determine its resolution.