  The fundamental plane of black hole activity (BHFP) describes the correlation
between radio luminosity ($L_R$), X-ray luminosity ($L_X$), and black hole
mass. It reflects a disc-jet connection. However, dependence of BHFP on various
physical properties of active galactic nuclei (AGNs) and host galaxies is
unclear, especially for low-luminosity AGNs, which is important for
understanding accretion physics in AGNs. Here we explore the dependence of BHFP
on radio loudness, Eddington ratio ($\lambda_E$), redshift, and galaxy star
formation properties at 0.1 < z < 4 for radio AGNs. Based on current deep and
large surveys, our studies extend to low luminosities and high redshifts. From
GOODS and COSMOS fields, we constructed a homogeneous radio AGN sample
including 208 objects. We divided the radio AGN sample into 141 radio-quiet
(RQ) AGNs and 67 radio-loud (RL) AGNs according to the radio loudness defined
by the $L_R$-to-$L_X$ ratio. This ratio shows a bimodal distribution that is
well described by two Gaussian models. Their cross point corresponds to a radio
loudness threshold of $\log (L_R/L_X) = -2.73$. RQ and RL AGNs show different
BHFPs, indicating a dependence of BHFP on the radio loudness. For both RQ and
RL AGNs, BHFP shows a dependence on $\lambda_E$, but no dependence on redshift.
BHFP shows a dependence on the galaxy star formation properties for RQ AGNs,
while for RL AGNs this dependence disappears. BHFP sheds important light on the
accretion physics and X-ray emission origins. X-ray emission of RQ AGNs
originates from a combination of advection-dominated accretion flow (ADAF) and
synchrotron radiation from jet at $0.01 < \lambda_E < 0.1$, and from jet at
$0.1 < \lambda_E < 1$. X-ray emission in RL AGNs originates from a combination
of ADAF and jet at $\lambda_E < 0.01$, from jet at $0.01 < \lambda_E < 0.1$,
and from a combination of standard thin disc and jet at $\lambda_E > 0.1$.
