On normal forms of gradient Ricci 4-solitons

In this note we analyze the normal form of the operator $\widehat{R}+\frac{1}{2}\widehat{H}$ of a gradient Ricci 4-soliton in Cao & Tran. In particular, we show that the curvature operator $\widehat{R}$ of the Koiso-Cao soliton inherits this normal form. By work of D. Johnson, this yields a normal form for the curvature operator of the Koiso-Cao soliton relative to the space of algebraic Kähler curvature operators.