For interpersonal situations, people often just want to be heard without any advice being given. A simple “I hear what you’re going through and I’m here if you need anything” is way better than finding ways to immediately offer advice, but I believe this applies to professional situations as well.
Often if I’m the person who is stuck I’m totally willing to let a new set of eyes offer all of the “obvious” things that have already tried, because while it’s usually just a rehash, there are occasionally “but what about” cases that haven’t been checked yet. And if we don’t find anything new, at least the helper is now up to speed on what’s already been covered.
Same here, with a slight variation. I usually say "my first thought is to try ..." or "the first thing that comes to mind for me is ..."
Depending on the situation I might also use the old "Have you tried ..." phrasing.
Branching way off topic, and paraphrasing Sun Tsu here, but he says to always give your opponent an out unless you plan to completely destroy them. Very rarely in business do you want to or need to destroy someone so I try to soften my suggestions.
The other mantra I repeat almost daily is 'do I want to be right or effective'. It feels great to tell someone an idea you know is 100% right, but does that mean they'll use it? At the end of the day I want to be effective. "Just" feels more like being right, than effective.
Speaking of phrases, early in my career the senior VP of my business unit was a nice, very sharp woman. One of the best executives I've ever met. Listening to her on calls was a master class on EQ and how to deal with people professionally. But, anytime she said 'help me understand...' you knew the hammer was coming.
A lot of ones I've seen is hand-wavey "be more empathetic" or "there are multiple intelligences" kind of advice.
I'd like to be able to listen in on conversations with challenging DO's and DON'Ts.
e.g. "Why do you think Rajesh reacted this way? What might James have said to elicit a more positive reaction, all other things being the same?"
Not so much intercultural communication as "here's how to give someone an out".
Usually it’s more along the lines of “for my own understanding, why can’t we just ship things as they’re ready instead of waiting for everything in the order to go in one box?”