Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group of $\mathbb{C}$-algebra automorphisms of $\mathcal{A}$ for low $\dim_\mathbb{C} \mathcal{A}$. It doesn't need to contain the actual computations (though that would be nice too), I just need a list with a few examples. I distinctly remember seeing such computations somewhere, but for the life of me I can't remember/find the source. For instance, it's not difficult to check that $\operatorname{Aut}_\mathbb{C}(\mathbb{C}[x]/(x^2)) \cong \mathbb{C}^\times$.
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1$\begingroup$ For a finite-dimensional algebra $A$, a computer can quickly compute the Lie algebra $\mathrm{Der}(A)$, which is the Lie algebra of $\mathrm{Aut}(A)$. This does not exactly determine $\mathrm{Aut}(A)$ but this is not very far. $\endgroup$– YCorCommented Jun 9, 2023 at 21:45
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$\begingroup$ @YCor: Thanks! Can you suggest software that can do this? $\endgroup$– M.G.Commented Jun 9, 2023 at 22:16
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$\begingroup$ I don't know (but this seems not hard to implement, e.g. in Python/SageMath, since a $n$-dim algebra is input by $n^3$ structure constants, and being a derivation consists in computing the set of solutions of a system of $n^3$ equations on $n^2$ variables, this system depending linearly on the "vector" of structure constants). $\endgroup$– YCorCommented Jun 9, 2023 at 22:24
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