Let $\mathfrak g$ be a Lie algebra over a field $k$. Everybody loves to study its automorphism group,
$$\mathrm{Aut}(\mathfrak g) := \{\alpha \in Aut_k(\mathfrak g): [\alpha(x),\alpha(y)] = \alpha([x,y]) \text{ for all } x,y \in \mathfrak g\}$$
which, if $\mathfrak g$ is sufficiently nice, can be an algebraic group over $k$, and for $k \in \{\mathbb R, \mathbb C\}$ might even be a Lie group.
But the following is also a group:
$$\mathrm{F}(\mathfrak g) := \{\phi\in Aut_k(\mathfrak g): [\phi(x),\phi(y)] = [x,y] \text{ for all } x,y \in \mathfrak g\}$$
Note that $-id \in \mathrm{F}(\mathfrak g)$ and hence $\mathrm{F}(\mathfrak g)$ is never trivial for $\mathfrak g \neq 0$ (unless $char(k)=2$, which we shall exclude from now on).
I wondered if it has been studied, and why I have never heard of it. Of course the suspicion is: Because it is "somewhat trivial" or "something known in disguise".
Indeed if we tried something like that for a unital associative $k$-algebra $R$,
$$\mathrm{F}(R) := \{\phi\in Aut_k(R): \phi(x) \cdot \phi(y) = x \cdot y \text{ for all } x,y \in R \}$$
it follows pretty easily that $\mathrm{F}(R) = \{\pm id\}$ is not interesting.
Well:
$\mathfrak g$ is abelian if and only if $\mathrm{F}(\mathfrak g) = \mathrm{Aut}(\mathfrak g)$.
Actually, all $ f \in \mathrm{F}(\mathfrak g) \cap \mathrm{Aut}(\mathfrak g)$ act as the identity on all commutators, hence on the derived subalgebra $[\mathfrak g, \mathfrak g]$. In particular, if $\mathfrak g$ is perfect, $\mathrm{F}(\mathfrak g) \cap \mathrm{Aut}(\mathfrak g) = \{id\}$.
Further, when restricted to $[\mathfrak g, \mathfrak g]$, all $\phi \in \mathrm{F}(\mathfrak g)$ commute with all $\alpha \in \mathrm{Aut}(\mathfrak g)$.(I realised my reason for this was flawed, and I am actually not sure about this.)A hands-on calculation shows that $\mathrm{F}(\mathfrak{sl}_2(k)) = \{\pm id\}$ and still I suspect that this is the case for a large class of Lie algebras.
Question 1: I suspect from the last point, using something like Jacobson density, or a Lie algebra version of von Neumann's bicommutant theorem, or something else that I just don't see, that for (semi)simple $\mathfrak g$, we have that $\mathrm{F}(\mathfrak g)$ contains only scalars, i.e. actually is as trivial as it can be, $\{\pm id\}$. Is that the case? What is the centraliser of $\mathrm{Aut}(\mathfrak g)$ in $\mathrm{Aut}_k(\mathfrak g)$ for (semi)simple $\mathfrak g$?
However, there is a lot of room between semisimple and abelian Lie algebras. E.g. for the non-abelian two dimensional Lie algebra $\mathfrak g$ with basis $x,y$ and $[y,x] =x$, written in that basis, we have
$$\mathrm{Aut}(\mathfrak g) = \{ \pmatrix{a&b\\0&1} :a \in k^\times, b \in k \}$$
$$\mathrm{F}(\mathfrak g) = \{ \pmatrix{a&b\\0&a^{-1}} :a \in k^\times, b \in k \}$$
in particular $\mathrm{F}(\mathfrak g) \cap \mathrm{Aut}(\mathfrak g) = \{ \pmatrix{1&b\\0&1} : b \in k \}$ is unipotent and (as mentioned before) acts trivially on the derived subalgebra $k \cdot x$; but also, $\mathrm{F}(\mathfrak g)$ and $\mathrm{Aut}(\mathfrak g)$ together generate the group $\{ \pmatrix{a&b\\0&c} :a, c \in k^\times, b \in k \}$ of all $k$-linear automorphisms of $\mathfrak g$ which respect the derived series.
Question 2: Does this "polarisation" phenomenon generalise to something interesting for a bigger class of Lie algebras, i.e. what can one say in general about the intersection $\mathrm{F}(\mathfrak g) \cap \mathrm{Aut}(\mathfrak g)$ and the product $\mathrm{F}(\mathfrak g) \cdot \mathrm{Aut}(\mathfrak g)$ in $\mathrm{Aut}_k(\mathfrak g)$?
Not an answer, but yes, this has been studied, at least on the Lie algebra level. Consider the Lie algebra of the Lie group ${\rm Aut}(\mathfrak{g})$, i.e., ${\rm Der}(\mathfrak{g})$ and similarly the Lie algebra $f(\mathfrak{g})$ of $F(\mathfrak{g})$. Then $f(\mathfrak{g})$ is a Lie algebra of generalized derivations. There is a lot of literature on such spaces. One of the most important references there is the following paper by Leger and Luks:
Leger, Luks: Generalized Derivations of Lie Algebras, Journal of Algebra ($2000$).
For applications see also our paper Post-Lie algebra structures on pairs of Lie algebras.