On a Generalized Mizuhara Construction

We describe the ideals for Mizuhara extensions and findsome necessary and sufficient conditions for the simplicity ofthe direct Mizuhara extension.Also, we study the Mizuhara constructionfor the matrix algebra and Burde algebras.We construct some various generalizationsof the Mizuhara construction and exhibit some examplesof the simple pre-Lie algebras that are obtained by thisconstruction; in particular, we construct the simple Witt doubles $ {\mathcal{A}}_{d} $ and $ {\mathcal{W}}_{d}({\mathcal{A}}) $ for a unital associative commutativealgebra $ {\mathcal{A}} $ with derivation  $ d $ .