The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor

In this article, we give a new proof of a result due to J. Kim, which states that the Ricci tensor of a gradient Ricci soliton with dimension $n\ge 4$ and harmonic Weyl tensor has at most three distinct eigenvalues. This result constitutes an essential step in the classification of such manifolds, originally established by J. Kim in dimension $4$ and subsequently extended to dimensions $n\ge 5$. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame.