On isometric stability of complemented subspaces of $L_p$

We show that Rudin-Plotkin isometry extension theorem in $L_p$ implies that when $X$ and $Y$ are isometric subspaces of $L_p$ and $p$ is not an even integer, $1\le p<\mathrm{\infty }$, then $X$ is complemented in $L_p$ if and only if $Y$ is; moreover the constants of complementation of $X$ and $Y$ are equal. We provide examples demonstrating that this fact fails when $p$ is an even integer larger than 2.