Automorphism Groups of Pre-Lie Witt Doubles

Let $ {\mathcal{A}} $ be a finite-dimensional associative commutative $ d $ -simple algebrabut not a field with nonzero derivation $ d $ over an algebraicallyclosed field $ F $ .We describe the automorphisms of the left-symmetricWitt doubles $ {\mathcal{A}}_{d} $ and $ {\mathcal{W}}_{d}({\mathcal{A}}) $ over $ F $ modulo the automorphisms of  $ {\mathcal{A}} $ almost commuting with  $ d $ .