Mishnah Berurah — משנה ברורה

On another garment, that is not allowed - Because there is a prohibition [in doing so], that he disgraces a טלית that's used for a mitzvah for no reason, unless the טלית is worn out, then it would be permitted to untie its' ציצית under any circumstance: [such as] if he wants to sell it to a gentile, or to make from the טלית another garment that doesn't have 4 corners, [in all those cases] it is permitted to take it [off]. But if he wants to sell the טלית to a Jew, it is forbidden to take [off] the ציצית, even if the other Jew can place [his own] ציצית [on it] himself, even here [in such a case] it is forbidden. And it is permitted to remove the ציצית and put in it's place nicer ones due to [it being an] enhancement of the mitzvah. Or if they (ie. the ציצית) were old and he removes them in order to put new ones. Or if one string severed and there was only left on it enough to loop around the other strings, even though the ציצית are still valid, nevertheless it is still permitted to remove it and place another whole [string] in it's place. And know further, that when the ציצית are whole and fitting to be placed in another garment [as opposed to if they only had enough to loop around the other strings], there are some from the later rabbinic authorities who wrote that one should be careful to undo the knots and their windings, and not to sever or tear them in order to not ruin them. And the sages already said "One should not pour out his well water when others are in need of it". And the Chayei Adam wrote that when it seems to a person that it will require great effort to untie them, [in such a case] it is permitted to sever them, and it doesn't [violate the prohibition] of "Do not destroy" since he is not doing it [simply] for the purpose of destroying. See there further.