Corollary 1 (Wedderburn’s theorem)

Let K be a commutative field, A a K-algebra, and $\mathfrak{r}$ the radical of A. Suppose that the K-algebra $A/\mathfrak{r}$ is absolutely semisimple.

a) Let $S_1$ and $S_2$ be subalgebras of A satisfying $A = S_1\oplus \mathfrak{r}= S_2\oplus \mathfrak{r}$. There exists a special automorphism of A transforming $S_1$ into $S_2$.

b) If $\mathfrak{r}$ is nilpotent, then there exists a subalgebra S of A satisfying A = $S\oplus \mathfrak{r}$.

This follows from Proposition 7 and Theorem 2 of VIII, p. 238.