# Left ideals of central simple algebra generated by symmetric element

Let $F$ be a field of characteristic $\neq 2$. Suppose $(A,\sigma)$ is a finite dimensional central simple algebra over a field $F$ with an orthogonal involution. Assume (A,σ) is non-split. I want to show that every left ideal is generated by a symmetric element. It appears as an exercise in 'The Book of involution', Ch.I. Can somebody provide hint for doing this?

As it is known that every left ideal of $A$ is of the form $Ae$ for some idempotent $e$. I was trying to 'modify' $e$, but not able to do it.

Thanks!

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Apply Artin-Wedderburn? –  Will Sawin Mar 8 '13 at 19:31