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How to create a shortcut or hot key to repeat the last action done in LyX to be as in MS Word F4 key?

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This does not exist. See Help > LyX Functions for a list of all of the functions. The reason this is difficult is because almost everything you do is considered an action. Moving the cursor left, entering a letter, etc. However, you can make a feature request here http://www.lyx.org/trac if you define what you mean by "last action". Here are a couple of options:

(1) Repeat the last command entered in the mini-buffer (alt-x).

(2) Repeat the last command of an appropriate set, e.g. {formatting text style, changing the properties of a figure}

(3) Do (2) where the appropriate set is defined by any action for which there is a shortcut in the user bind file (well, excluding the basic commands, see below).

I think (2) is what Word probably does. But this is difficult and would require a lot of work to implement from scratch. I just thought of (3) now and that seems the most interesting to me but I haven't thought through it yet. I like it because it can allow customization of "last action" by the user. All of the commands (like move the cursor left) are also defined indirectly by the bind file but it might be interesting to only take the commands that are specific to each bind file (e.g. exclude any "includes" in the bind files. And then allow the user to add those back in.).

If you do file an enhancement request, please link back to this question in that request and please link to that request in this question.

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