On the real forms of the exceptional Lie algebra $ \mathfrak{e}_6$ and their Satake diagrams

TL;DR

This paper systematically develops Satake diagrams for the real forms of the exceptional Lie algebra E6, using algebraic constructions involving Albert and paraoctonion algebras, enhancing understanding of their structure.

Contribution

It provides explicit Satake diagrams for three real forms of E6, employing novel algebraic constructions with Albert and paraoctonion algebras.

Findings

Explicit Satake diagrams for $ rak{e}_{6,-26}$, $ rak{e}_{6,-14}$, and $ rak{e}_{6,2}$.

Construction methods using Albert algebra and paraoctonion algebras.

Insight into the structure of real forms of E6.

Abstract

Satake diagrams of the real forms $ \mathfrak{e}_{6,-26}$, $ \mathfrak{e}_{6,-14}$ and $ \mathfrak{e}_{6,2}$ are carefully developed. The first real form is constructed with an Albert algebra and the other ones by using the two paraoctonion algebras and certain symmetric construction of the magic square.