Tits-Kantor-Koecher construction for Jordan algebra of symmetric bilinear form

Question

Suppose $f$ is a nondegenerate, symmetric bilinear form on a vector space $V$ over a field $F$. Then $J = V + F\cdot 1$ is a unital Jordan algebra (known as the "spin factor" Jordan algebra when $F = \mathbb{R}$ and $V = \mathbb{R}^n$). We can construct the Lie algebra $TKK(J) = L = L_{-1} + L_0 + L_1$ via the Tits-Kantor-Koecher construction where $L_1 \cong L_{-1} \cong J$. I have seen several papers reference the fact that $TKK(J) \cong K(R, *)$, the Lie algebra of skew-symmetric elements in an associative algebra $R$ with involution *. Can someone direct me to a proof of this fact? In particular, I would like to know what is $R$ and I'd like a proof when $V$ is not necessarily finite dimensional.

Answer 1

This is contained in Jacobson's Blue Book (Structure and Representations of Jordan Algebras, AMS Colloquium Publications, 1968), as Exercise 1 on p. 342, for arbitrary fields, and with no assumptions on the dimension of $V$.

More precisely, let $W$ be the $3$-dimensional vector space with bilinear form $g$ given by the matrix $\left( \begin{smallmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{smallmatrix} \right)$, and consider the orthogonal sum $h := f \perp g$ on $U := V \oplus W$. Then $TKK(J)$ is isomorphic to the Lie algebra of linear transformations of $U$ that are skew w.r.t. $h$.