# On Kac's Jordan superalgebra

## TL;DR

This paper characterizes the automorphism group of Kac's ten-dimensional Jordan superalgebra, using it to classify its twisted forms and gradings more simply and comprehensively.

## Contribution

It identifies the automorphism group as a specific semidirect product, extending and simplifying the classification of twisted forms and gradings of the superalgebra.

## Key findings

- Automorphism group is isomorphic to a semidirect product of SL2×SL2 and C2.
- Provides a unified framework for classifying twisted forms.
- Simplifies previous classification results.

## Abstract

The group-scheme of automorphisms of the ten-dimensional exceptional Kac's Jordan superalgebra is shown to be isomorphic to the semidirect product of the direct product of two copies of SL2 by the constant group scheme C2. This is used to revisit, extend, and simplify, known results on the classification of the twisted forms of this superalgebra and of its gradings.