Be sympathetic with the other person's ideas and desires
The core idea of the chapters in [@carnegie2010How to win friends & influence people] is that if someone is angry at us for something we have done or said, instead of immediately fighting back, we should look at the situation from the other's perspective. This has already been discussed in Try honestly to see things from the other person's point of view, Show respect for the other person's opinions. Never say You're wrong, Make the other person feel important - and do it sincerely, and partially in Be a good listener. Encourage others to talk about themselves.
The examples range from someone mad because the author gave the wrong birth place of a writer on the radio, the singers, to business salesmen. One good practice, even by today standards, would be to write an angry reply and never send it. Sleep it off, and the following day re-write the e-mail in much more amenable terms.
Literature note on The Leader in You - Dale Carnegie
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