We define an algebraic group to be solvable if it has a filtration where the successive quotients are abelian. The Lie-Klochin theorem says that every smooth solvable connected group admits an embedding into the Borel subgroup of $GL_n$. But the Borel subgroup o $GL_n$ has a filtration where the successive quotients are $\mathbb{G}_m's$ and $\mathbb{G}_a's$.
Therefore it seems like there are really no examples of abelian linear algebraic groups other than $\mathbb{G}_m's$ and $\mathbb{G}_a's$ or tori and finite abelian groups. Unless the composition factors for a linear algebraic group need not be unique, unlike the case for finite groups.
Are the composition factors for a linear algebraic group unique, and if not, then what are some more interesting examples of abelian linear algebraic groups?