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.