  Astrophysical black holes are ideal laboratories for testing Einstein's
theory of general relativity in the strong field regime. In this manuscript, we
present an analysis of Suzaku and NuSTAR spectra of the black hole binary
Cygnus X-1 using RELXILL_NK. Unlike our previous study on Cygnus X-1 with
NuSTAR data, here we are able to constrain the Johannsen deformation parameter
$\alpha_{13}$. However, despite the high energy resolution near the iron line
provided by Suzaku, our constraints on the Kerr metric from Cygnus X-1 are not
very stringent in comparison with those that have been obtained from other
sources, confirming that Cygnus X-1 is quite a complicated source.
