Lie Groups and Lie algebras related to Jordan algebras

Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$.


In the paper “The Capelli Identity, Tube Domains, and the Generalized Laplace Transform

Jacobson (J) has associated two Lie algebras to $J$. For any $x in J$, let $L(x)$ denote the operator of left multiplication by $x$. Let
$mathfrak{p} = {L(x): xin J}$ and let $mathfrak{t} = (mathfrak{p}, mathfrak{p})$. Then $mathfrak{t}$ is the Lie algebra of the “automorphism
group” of $J$. Let $$mathfrak{g}=mathfrak{t}+mathfrak{p}$$
Then $mathfrak{g}$ is a reductive Lie algebra and above is the Cartan decomposition
of $mathfrak{g}$.

The paper (J) mentioned is “Some groups of Transformations defined by Jordan Algebras I.


In the note “Lecture notes in mathematics: an elementary approach to bounded symmetric domains“.

  1. The binary Lie algebras and symmetric Lie algebras were defined on Page 9 and Page 35.

  2. The Meyberg theorem on the relation to Jordan algebra was proved on Page 19.

  3. The construction of Lie algebras can be found in Page 80 and Page 46.

As far as I see, the resulting Lie algebra $mathfrak{g}=mathfrak{T}oplus Joplus J$ has two copies of $J$.


In the book “Jordan algebras and algebraic groups“.

  1. The $J$-structure was introduced in section 1 (it is equivalent to a Jordan algebra structure for characteristics $neq 2$, section 6). There is a structure group $G$ associated to it.

  2. In section 11, we pick some idempotent $a$ to define $S_a$ and give the classification of simple $J$.-structure. It seems to be related to the exceptional group $E_6$ on Page 115.


My question is,

what is the relation between them? Do we have a list of computation of $mathfrak{g}$ for real simple $J$‘s? Besides, what is the geometry of it? For example, is it the isotopic group of the symmetric domain?