Possible typo in this paper differentiating a matrix product

Question

I'm reading this paper and it seems to me like there's a typo at the top of page 53. Here's the problematic paragraph:

The possible typo is on the third line.

1. $S = R^\dagger \lambda R$
2. $dS = dR^\dagger \lambda R + R^\dagger d\lambda R + R^\dagger \lambda dR$ by the chain rule
3. But the expression they give is $dS = R^\dagger (d\lambda + [\lambda, R^\dagger dR])R$ = $R^\dagger d\lambda R + R^\dagger \lambda R^\dagger dR R - R^\dagger R^\dagger dR \lambda R$
4. We know $R$ is a rotation matrix so $R^\dagger R = I$. Differentiating this yields $dR^\dagger R + R^\dagger dR = 0$, but even if we use this in #3, we don't get the result in #2.

It seems to me that there's a typo here and the correct expression should be $dS = R^\dagger (d\lambda + [\lambda, dR R^\dagger ])R$.

Is this the case? If so, the rest of the derivation seems to be in jeopardy, since the "typo" is propagated.