Should one avoid using the phrase "note that" in math writings? I always thought that the phrase "note that" in a mathematical writing is simply something you say to express that what comes after "note that" is a fact. Due to its neutrality and versatility, I use it a lot, without even thinking the possible sentiments that might possibly make some people get me wrong. Moreover, many times we have to prevent symbols to appear directly after a comma or a period, so "note that" is convenient in this aspect too. 

However, some incidents lead me to presume that perhaps those readers who are not that mathematically experienced could take "note that" as an imperative or even a condescending sign? This is much to my surprise. So, if I am writing something to someone that is senior than me in academia and that is not that mathematically experienced (say an applied engineering scholar whose math background is all from the school curriculum), would it be suggested that I avoid using "note that"? Thank you.
"Note that" is a useful rhetorical device in mathematical writing. Its role is to signal to the reader that the statement following it is more noteworthy than other statements, so the reader should, quite literally, take note. While @JeffE sarcastically argues in a comment1 that it must never be used, I disagree; a judicious use of "note that" (in contrast to using it either always as in Jeff's comment, or never) helps to effectively convey nuances of meaning to the reader that are difficult to convey otherwise. This is an example of taking advantage of the English language in mathematical writing. In essence, "note that" punctuates the flow of the text and helps the reader process and make sense of the large amount of information that a typically dense mathematical text often contains, and discern which parts of it are more important and which are less. 

Of course, as with any other form of expression, "note that" can be used effectively, or (as JeffE's comment cleverly illustrates) abused horribly.

Now to address your questions:


  some incidents lead me to presume that perhaps those readers who are not that mathematically experienced could take "note that" as an imperative or even a condescending sign?


I really can't say whether "note that" will come across as condescending to those who aren't used to its usage in mathematical writing, but I suppose that's possible (and I'd be curious to hear more about those "incidents"). "Note that" is in fact an imperative, so it makes sense that it will be understood as such by anyone, mathematically experienced or not.


  if I am writing something to someone that is senior than me in academia and that is not that mathematically experienced [...] would it be suggested that I avoid using "note that"? 


My recommendation is that when you are writing to someone who is less mathematically experienced, just like you need to be mindful that you must adapt the mathematical terminology you're using to your reader's expectations and knowledge, it probably makes sense to also adapt the style of writing accordingly to take into account cultural differences across different disciplines. It is probably true that mathematicians are more used to the way "note that" sounds and is typically used, so one needs to tread a bit more carefully with this device when speaking to a non-mathematician. I wouldn't say you should never use it however, just maybe pay a bit more attention to when and how much you use it and give some thought to whether each usage makes sense in the context of how it's likely to come across.



1 JeffE's comment I was referring to, which was moved to chat, said: 'Note that “note that” is simply redundant. Note that omitting it entirely does not change the meaning of the sentence. Note that it’s just a bad writing habit. Note that you don’t need it.'