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

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.

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Is $R$ the Clifford algebra of the form you start with? –  Bruce Westbury Apr 27 '13 at 6:58