Classification of 5-dimensional restricted Lie algebras over perfect fields, I

The purpose of this paper is to classify all $p$-nilpotent restricted Lie algebras of dimension 5 over perfect fields of characteristic $p\ge 5$. Our work builds upon the recent work of Schneider and Usefi on the classification of $p$-nilpotent restricted Lie algebras of dimension up to 4 over perfect fields of characteristic $p$. The method we use here to classify $p$-nilpotent restricted Lie algebras is the analogue of Skjelbred-Sund method for classifying nilpotent Lie algebras.