There are only a couple Lakota Sioux potters that I'm aware of who make hand-made pottery: Elmer Red Starr and his nephew, Norman Red Star. Both use Santa Clara Pueblo clays, methods and styles with their own Sioux sgraffito designs.
There is a factory in Rapid City, SD, that turns out Sioux pottery using red clay dug in the Black Hills and white commercial clay from Kentucky. Other than the Sioux who do the actual designs, the operation is a typical factory churning out thousands of copies of the same stuff, kinda like the "greenware" turned out by some artisans in Colorado and New Mexico. Nothing unique, nothing to differentiate one piece from any other, except maybe in the designs and their placement.
When the Puebloans make pottery, almost all dig and process their own clay, make their own pots by hand using techiques that have changed little in several hundred years, decorate with designs that are also several hundred to several thousand years old, and they have prayers to utter during every step of the process. If you are sensitive enough to feel the Clay Mother, you'll feel the spirit of those prayers in that pottery. That's why so much Puebloan pottery has spirit lines in the decorations: to allow Spirit/Clay Mother to flow freely in and out of the piece. But you won't get that feel from greenware (usually made in a slip-cast form: pour in liquid clay, shake to mix and let dry before sanding and decorating) or any other pottery produced in a factory setting, no matter how many spirit lines it shows.
That said, the triangular symbols used by the Seven Indian Tribes of the Lakota depict their connection with the nature surrounding them and with important events in tribal history. These symbols flowed all through the intricate designs of their traditional bead work and decorated their clothing, weapons, utensils and pottery. Sioux symbols alone, or in combination, express the spirit and culture of the Lakota American people.
Standing up to six feet tall and weighing up to 2,000 lbs, Tatonka was worshipped as a sacred animal, its spirit prayed to before every hunt. Tatonka (the American bison) is essentially the heart and soul of the Lakota Sioux. From spoons to weapons to clothing to tipis, they were able to get virtually everything they needed for their existence from Tatonka. Every part of the buffalo not used for food was used for something else (including the bones, hoofs, horns, hides and internal organs). Nothing of the sacred Tatonka was wasted. Even the dung was saved and used for fuel.
Most of the Plains tribes were similar in their reverence for the buffalo and that symbology has made its way across virtually the entire gamut of today's Native American artwork, and not just with the Plains tribes.
The problem with "tradition" is that it was hard for the Sioux to transport materials in fragile containers, like pottery, on horseback. It was much easier using baskets. So they had no real pottery tradition, what there is has developed over the last few decades.