Starting from $X$ one can construct a sequence $(X_0,X_1,\dots)$ of groupoid objects in derived spaces, all having $X$ as space of objects (I am writing "space" in order not to specify the geometric context I am working with, but I actually mean scheme):

1. $X_0$ is the pair groupoid $X\times X$ of $X$. The Lie algebroid of this groupoid is $TX$.

2. $X_1=X\times^h_{X_0}X$ is the derived self-intersection of the diagonal in $X_0$ (i.e. the derived loop space of $X$). The Lie algebroid of this groupoid is $TX[-1]$ with the Atiyah Lie bracket. It happens to be a Lie algebra (because the derived loop space is actually a group).

...etc... In general $X_{n+1}=X\times^h_{X_n}X$.

Another way to think about the relation between $TX$ and $TX[-1]$ is to say that $TX[-1]=\Omega_0(TX)$. Hence  $TX[-1]$ is a group in Lie algebroids, and thus must be a Lie algebra. One actually sees this way that the Lie algebroid of $X_2$ is going to be $\Omega_0(TX[-1])$, and thus is a group in Lie algebras... hence it must be an abelian Lie algebra.

The hierarchy Lie algebroid -> Lie algebra -> abelian Lie algebra -> ... one gets is analogous to the following one we get in algebraic topology: set -> group -> abelian group -> ...

The upshot is that $X_0$ is a groupoid, $X_1$ is a group, and $X_n$'s are abelian groups for all $n\geq2$.

Starting from $X$ one can construct a sequence $(X_0,X_1,\dots)$ of groupoid objects in spaces, all having $X$ as space of objects (I am writing "space" in order not to specify the geometric context I am working with, but I actually mean "derived scheme"):

1. $X_0$ is the pair groupoid $X\times X$ of $X$. The Lie algebroid of this groupoid is $TX$.

2. $X_1=X\times^h_{X_0}X$ is the derived self-intersection of the diagonal in $X_0$ (i.e. the derived loop space of $X$). The Lie algebroid of this groupoid is $TX[-1]$ with the Atiyah Lie bracket. It happens to be a Lie algebra (because the derived loop space is actually a group).

3. In general $X_{n+1}=X\times^h_{X_n}X$ and its Lie algebroid $\mathfrak g_{n+1}$ is $TX[-n-1]$, and is an abelian Lie algebra as soon as $n\geq1$.

In other words, we have that $\mathfrak g_n=\Omega^n_0(TX)=TX[-n]$. Hence, in particular $TX[-1]$ is a group in Lie algebroids, and thus must be a Lie algebra. You get that the Atiyah-Kapranov Lie bracket on $TX[-1]$ is completely dertmined by the Lie algebroid structure on $TX$.

One also gets that the Lie algebroid $\mathfrak g_2$ of $X_2$ is going to be $\Omega_0(TX[-1])$, and thus is a group in Lie algebras... hence it must be an abelian Lie algebra.

The hierarchy Lie algebroid -> Lie algebra -> abelian Lie algebra -> ... one gets is analogous to the following one we get in algebraic topology: set -> group -> abelian group -> ...