Nonexcellence of certain field extensions

Consider towers of fields $F_1\subset F_2\subset F_3$, where $F_3/F_2$ is a quadratic extension and $F_2/F_1$ is an extension, which is either quadratic, or of odd degree, or purely transcendental of degree 1. We construct numerous examples of the above types such that the extension $F_3/F_1$ is not $4$-excellent. Also we show that if $k$ is a field, $\operatorname{char}k\ne2$ and $l/k$ is an arbitrary field extension of forth degree, then there exists a field extension $F/k$ such that the forth degree extension $lF/F$ is not 4-excellent.