Note on identifying four-dimensional vacuum solutions from Weyl invariants

The diffeomorphism covariance is a fundamental property of general relativity that leads to the fact that the same solution of the Einstein equation can be given in completely distinct forms in different coordinate systems. Distinguishing or identifying two metrics as solutions of the Einstein equation is particularly challenging. In a recent paper [Lu and Mao, Four-dimensional stationary algebraically special solutions and soft hairs, arXiv:2503.14586.], it was proposed to apply the relations of different Weyl invariants to distinguish solutions. In this paper, we present a complementary application of the Weyl invariants. We verify from Weyl invariants that two metrics with completely different forms are the same solution. We also present the coordinates transformation that connects the two metrics.