# Infinitesimal left actions are Lie algebra ANTIhomomorphisms?

Before I ask the question I clear up some notation. Throughout this question $M$ will be a smooth $n$ dimensional manifold, $G$ a $k$ dimensional Lie group and $G\times M\rightarrow M$ a smooth left action of $G$ on $M$.

If $X\in\mathfrak X(M)$ is a smooth vector field on $M$, its flow will be denoted as $\phi^X$. The effect of flowing for "time" $t$ from the point $x\in M$ is $\phi^X_t(x)$.

The Lie derivative of an arbitrary tensor field $T$ along $X$ is defined as $$\mathcal L_X T=\frac{d}{dt}(\phi^X_t)^\ast T,$$ where this derivative is evaluated at $t=0$, and the upper asterisk denotes pullback. In particular, for a vector field $Y$, this means $$\mathcal L_XY|_x=\frac{d}{dt}T\phi^X_{-t}(Y|_{\phi^X_t(x)}),$$ and this derivative is evaluated at $t=0$, and $T$ is the tangent functor. As far as I am aware, this is the standard convention.

Using these conventions, it can be derived, that if $A,B\in\mathfrak g$ are Lie algebra elements, then $$\text{ad}_AB=\frac{d^2}{dsdt}\exp(sA)\exp(tB)\exp(-sA)=[A,B],$$ and this derivative is evaluated at $s,t=0$.

If $A\in\mathfrak g$, then we define the infinitesimal transformation associated to $A$ as a smooth vector field $X_A\in\mathfrak X(M)$ defined by $$X_A|_x=\frac{d}{dt}\exp(tA)x,$$ where this derivative is evaluated at $t=0$, and $\exp(tA)x$ is the left action of $\exp(tA)$ on the point $x\in M$.

I am interested in the nature of the map $A\mapsto X_A$. Linearity is easy to see. I want to show how $X_{[A,B]}$ and $[X_A,X_B]$ are related.

My attempt at a derivation is as follows: $$X_{[A,B]}=\frac{d}{dsdt}\exp(sA)\exp(tB)\exp(-sA)x=\frac{d}{ds}\frac{d}{dt}\phi^A_s\circ\phi^B_t\circ\phi^A_{-s}(x) \\ =\frac{d}{ds}T\phi^A_s\left(\frac{d}{dt}\phi^B_t(\phi^A_{-s}(x))\right)=\frac{d}{ds}T\phi^A_s(X^B|_{\phi^A_{-s}(x)})=-\mathcal L_{X_A}X_B|_x=-[X_A,X_B]|_x.$$ Here all derivatives are evaluated at 0 and $\phi^A_s\equiv\phi^{X_A}_s$.

So it seems that the map $A\mapsto X_A$ is a Lie algebra antihomomorphism. This seems to be contrary to expectation, especially as $\text{ad}_AB=[A,B]$, and this result should essentially be analogous to the relationship between $\text{ad}$ and $[\cdot,\cdot]$.

Is my derivation correct? Is this a bug or a feature? If I'm right, is there any particular reason why this is an antihomomorhpism?