Additive group of ring of Witt vectors inherits algebraic group structure

For ring of Witt vectors
Suppose $$K$$ is a field and $$p$$ is a prime number. Suppose $$W$$ denotes the ring of Witt vectors over $$K$$ for the prime $$p$$. Then, the additive group of $$W$$ naturally inherits the structure of an algebraic group over $$K$$, with the algebraic group structure arising from the fact that the operations are defined by formulas with coordinates in $$K$$, and that these formulas work to give a group in any field extension of $$K$$.

For ring of Witt vectors truncated to finite length
Suppose $$K$$ is a field, $$p$$ is a prime number, and $$d$$ is a natural number. Consider the ring of truncated Witt vectors of length $$d$$, obtained by projecting the ring of Witt vectors to the first $$d$$ coordinates. The additive group of this ring is an algebraic group over $$K$$.

Related facts

 * Unipotent linear algebraic group structure of additive group of ring of Witt vectors of length two