 above would be equivalent to the Jacobi identify for $n^{(3)}$ interpreted as structure constants if the terms $m_i^{(4)}$ were absent. The presence of $m_i^{(4)}$ is the evidence that $n^{(3)}$ cannot be used as structure constants.

Often the relation between YM and GR indicated by the double-copy procedure is symbolically represented as (YM)${}^2 = $(GR). This expression is actually misleading, because neither the color factors or the propagators are squared in the gravity theory. Instead, the identification of color factors with kinematic factors (e.g. $f_{abc}$ with $n_{\lam\m\n}(p, q, -(p+q))$ for tree-level 3-point amplitudes) identifies YM directly with GR. It is more appropriate to use (YM)${}^{\prime}$ = GR as the symbolic representation of this connection. The prime on (YM) indicates the modification of YM theory by the replacement of color factors by kinematic factors.

Since the color factors are composed of structure constants of the gauge group, we are naturally led to consider the possibility of gauge symmetries with structure constants involving kinematic factors. This was precisely what we did in Sec.\[GaugeSymm\], which led to new formulations of gravity theories as generalized YM theories in Sec.\[YM\]

Apparently, the hope for a direct matching between structure constants and the kinematic factors is too naive. First, as the 3-point amplitude is defined on-shell, the structure constant can be different from $n_{\lam\m\n}(p, q, -(p+q))$ when it is off-shell. Secondly, even if the 3-point amplitudes agree with structure constant, it is not clear if higher-point amplitudes will automatically agree with the corresponding color factors, as there will be different on-shell conditions at work. (In fact, eq.(\[nn+m\]) says that the structure constants for 4-point amplitudes are not to be given by the structure constants for 3-point amplitudes due to the extra term $m^{(4)}_i$.) In general, as the BCJ duality only holds on-shell, the correspondence between structure constants and kinematic factors in