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The phrase "are firmly embedded" is correct and usable in written English
It can be used to describe something that is deeply ingrained or established within a particular context or system. Example: "The values of integrity and respect are firmly embedded in our company culture."
Exact(19)
Today they are firmly embedded in advertising history.
The intrinsic proteins, as their name implies, are firmly embedded within the phospholipid bilayer.
Uniform TiC nanoparticles (NPs) are firmly embedded into the CNFs matrix.
But the music industry is a cruel one and, five years on, Arctic Monkeys have just celebrated their fourth consecutive No 1 album and are firmly embedded in the cultural landscape.
By using this technique, CNTs are firmly embedded into the Ni micro wire.
The CNT roots are firmly embedded in the substrate, and no contamination is induced.
Similar(41)
It is firmly embedded in a community.
Her background is firmly embedded in Israel's right wing.
The empire was firmly embedded in the intellect of its subjects.
Only if secular government was firmly embedded in the Christian faith could its democratic institutions survive.
However, with Japanese news media breathlessly reporting the latest drop, the Nikkei is firmly embedded in the popular consciousness.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com