On left nilpotent skew braces of class 2$2$ 2

The main objective of this article is to present a detailed structure of left nilpotent skew braces upper B B B of class 2 $2$ 2 , i.e. skew braces with upper B 3 equals 0 B³ = 0 B 3 = 0 . We prove that if upper B B B is of nilpotent type, then upper B B B is centrally nilpotent. In fact, we show that upper B B B is right nilpotent of class at most 2 plus mr $2+mr$ 2 + m r , i.e. upper B left parenthesis 2 plus mr plus 1 right parenthesis equals 0 B⁽²⁺ᵐʳ⁺¹⁾ = 0 B ( 2 + m r + 1 ) </mml:m