The condition of being embroiled in something; an imbroglio or entanglement(noun)
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Use "embroilment" in a sentence
"Our noun "embroilment," which also entered the language in the early 17th century, comes from the same source."
"The most difficult task has been persuading the arts establishment that this is work to be taken seriously by art collectors and your regular visitor to Tate Modern, that it somehow transcends what we instinctively fear is an art contaminated by its embroilment with other disciplines."
"Also, they were critical of Mr. Vekselberg's public shyness and his embroilment in a criminal investigation."