This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along a smooth plane cubic or not. I believe that the answer shoud be "no" by Matsushima's criterion saying that if the automorphism group of the variety is not reductive then there is no such metric.

I was thinking that the following might be a (sketch of) proof: The automorphisms of $\mathbb{P}^3$ fixing a plane (say $x_3=0$) are of the form
$$\begin{pmatrix}
* & * & * & * \\
* & * & * & * \\
* & * & * & * \\
0 & 0 & 0 & *
\end{pmatrix} $$

This group has projective dimension 12 and since 9 points determines a cubic in $\mathbb{P}^2$ we have that the automorphism group of the blow-up is 3-dimensional. On the other hand, the 3-dimensional unipotent subgroup
$$\begin{pmatrix}
1 & 0 & 0 & * \\
0 & 1 & 0 & * \\
0 & 0 & 1 & * \\
0 & 0 & 0 & 1
\end{pmatrix} $$
acts trivially on the plane $x_3=0$, so in particular fixes the 9 points determining the cubic and hence lifts to the automorphism group of the blow-up. Since they have the same dimension (but the latter might be non connected), the automorphism group if a finite extension of a unipotent group, hence not reductive.

Am I right? Sorry for being sketchy! Thanks a lot in advance for any comment, I just started to introduce myself to the subject.

PS: By the way, does the fact that $\operatorname{Aut}(X)$ is reductive implies that the connected component of the identity $\operatorname{Aut}^0(X)$ is reductive as well?