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This study reveals that if a symmetric multilinear algebra is weakly nil, it becomes Engel, indicating how it connects to the Jacobian theorem.
If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed by Dotsenko.
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Dmitri Piontkovski (2025) studied this question.
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