The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:^{1}

**1.1. Lemma.**

If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ is any symmetric two tensor, then $$ \delta\left[\operatorname{DEin}\left({ }^{(4)} g\right) \cdot ^{4} h\right]=0 $$ where $\delta=\delta_{(4) g}$ is the divergence with respect to ${ }^{(4)} g$.

They gave the following proof:

The contracted Bianchi identities assert that $\delta \operatorname{Ein}\left({ }^{(4)} g\right)=0$. Differentiation gives the identity $$ **\left[\text { D } \delta\left({ }^{(4)} g\right) \cdot ^{4}h\right] \cdot \operatorname{Ein}\left({ }^{(4)} g\right)+\delta\left[D \operatorname{Ein}\left({ }^{(4)} g\right) \cdot^{4} h\right]=0** $$ where $\delta\left({ }^{(4)} g\right)=\delta_{(4)}$ indicates the functional dependence of $\delta$ on ${ }^{(4)} g$, and $\left[\mathrm{D} \delta\left({ }^{(4)} g\right) \cdot ^{4} \mathrm{h}\right]$ is the linearized divergence operator acting on Ein $\left({ }^{(4)} g\right)$. The lemma follows since $\operatorname{Ein}\left({ }^{(4)} g\right)=0$.

Sadly, I cannot see how they found the identity in the second sentence of the proof. Could you give me some hints/suggestions on how to derive the identity in the second sentence of Lemma 1.1?

Any help/suggestions would be highly appreciated. Thanks so much.

^{1}*Fischer, Arthur E.; Marsden, Jerrold E.; Moncrief, Vincent*, **The structure of the space of solutions of Einstein’s equations. I: One Killing field**, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 33, 147-194 (1980); eudml. ZBL0454.53044, MR605194.

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