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### Published volumes

##### 1936-1944
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4
Volume 62, no. 1 (2021), pp. 123–139

### Abstract

After the classification of simple Lie algebras over a field of characteristic $p > 3$, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie $2$-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie $2$-algebras with toral rank odd and furthermore that the simple contragredient Lie $2$-algebra $G(F_{4, a})$ of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of $G(F_{4, a})$.