Is there an even integer $n$ such that $S^n$ can be topologically embedded in its unit tangent bundle $$T^{1} S^n=\{(x,y)\in S^n \times S^n \mid x.y=0\}$$ What about if we pose the extra condition that the image of $S^n$ be a retract of the unit tangent bundle?

Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question:

For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?

Is there an even integer $n$ such that $S^n$ can be embedded in its unit tangent bundle $$T^{1} S^n=\{(x,y)\in S^n \times S^n \mid x.y=0\}$$ What about if we pose the extra condition that the image of $S^n$ is a retract of the unit tangent bundle?

Is there an even integer $n$ such that $S^n$ can be topologically embedded in its unit tangent bundle $$T^{1} S^n=\{(x,y)\in S^n \times S^n \mid x.y=0\}$$ What about if we pose the extra condition that the image of $S^n$ be a retract of the unit tangent bundle?