I would like to knit a cardigan in stockinette alternated with lace on a Brother standard gauge. Would I need to adjust the tension for the lace part so that it is not wider than the stockinette parts? Thank you!
Almost certainly, lace in generally a looser fabric than stockinette. A tension swatch or two will give the best indication.
I made a sweater with the body in stockinette the sleeves only in lace, and yes, you will have to knit a completely different gauge based on your swatch. However, if you include lace into one piece, it might create a texture that you like if you maintain the same gauge. The best you can do is experiment until you like what you see!
Make 2 swatches, one in the lace pattern and one in stockinette. If you are using a small amount of lace it won't make any difference, but if you are using a large amount of lace then you should take the measurements of the two and average them. If however you can also make a swatch of the amount of stockinette and lace together and use the average to get your stitch and row gauge. Make several measurements and average them to get your stitch gauge.
From a recent project, for my pattern, going 2 whole numbers lower for the lace than stockinette worked well for my pattern. I got about the same width for both stockinette and lace. That may be helpful for you as a starting point for your swatching.