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Solvability of derivation Lie algebras of local finite-dimensional commutative algebras

The answer is NO. Consider for instance the commutative algebra $A$ having the elements $a_0,a_1,\ldots,a_n$ as a $k$-basis such that $a_0$ is the unity of $A$ and $a_ia_j=0$ for all $i,j=1,\ldots,n$. ...
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