On normal forms of gradient Ricci 4-solitons

In this note we analyze the normal form of the operator $\hat{R} + \frac{1}{2}\hat{H}$ of a gradient Ricci 4-soliton in Cao & Tran. In particular, we show that the curvature operator $\hat{R}$ 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 K\"ahler curvature operators.