Use lag operators with interaction term, is it correct?

I'm sorry I make reference to Stata here, but it's just to illustrate the point, the question is about whether it is correct or not using lagged variables in interaction terms.

I have a linear model that I am estimating using FE and RE, in which my explanatory (continuous) variable is lagged, something like this:

$$y = \alpha + \beta x_{t-1} + \epsilon$$

Stata: xtregar y L.variable, fe/re

For my analysis, I need to include a categorical interaction term. Stata, by using the graphical interface, does not allow me to use lags with my continuous variable, but it does give me results if I add 'L' manually, and these are very different to those without the L.

What Stata's graphical interface allows me to do is:

xtregar y L.variable dummy#c.variable i.dummy, fe


And what I think I need is:

xtregar y L.variable dummy#cL.variable i.dummy, fe


I am confused since I imagine that if adding lags was OK, Stata would allow me to do so, wouldn't it?

1. Is it correct to add lags to an interaction term?
2. In case of that the answer to (1) is no: If I remove the variable (L.variable), is it ok to add lags to the interaction term?

A related question:

What to do if the variable is not significant but the interaction term is?