http://revach.net/article.php?id=3267

Section: Taryag Mitzvos   Category: Minchas Chinuch
Minchas Chinuch - When Left Is Right
The Mishnah states that a right-handed person who writes on Shabbos with his left hand is Patur because a person who does a Melachah in an unconventional fashion is Patur since he is not doing the Melachah in the way that it was done in the Mishkan. What is the Halachah if a person writes a Sefer Torah or a Get with his left hand? Is the Sefer Torah Kosher?

The Rema says that the Sefer Torah or Get is Kosher. Even though he is Patur on Shabbos, that is because it was done unconventionally, not because it is not considered writing.  Therefore it is considered writing with respect to a Sefer Torah or a Get. However, the Bais Yosef disagrees and he says that if it was considered a writing for a Sefer Torah, one would be Chayav for writing in that fashion on Shabbos. The Minchas Chinuch answers for the Rema that maybe it is considered writing for a Sefer Torah.   Therefore if one wrote a Sefer Torah with his left hand on Shabbos he would be Chayav, since it is considered a writing for a Sefer Torah it is also considered a writing for Shabbos.  However if a person writes anything else with his left hand other, than a Sefer Torah on Shabbos, he would be Patur.