Sentence examples for not however from inspiring English sources

"not however" is a correct and usable phrase in written English.
It is used to introduce a contrasting idea or statement to the previous sentence. Example: John was not, however, willing to give up on his dream of becoming a lawyer. In this sentence, the phrase "not however" is used to contrast John's unwillingness to give up on his dream with the previous statement that he was not willing to compromise his principles.