I am taking the definition that a solvable algebraic group $G$ is split if it has a composition series: $$1=G_0\lhd G_1\lhd \dots \lhd G_n=G,$$ such that each quotient $G_i/G_{i-1}$ is isomorphic to either $\Bbb G_m$ or $\Bbb G_a$.
Then, I've read that
- a general algebraic group is split if it has a split Borel subgroup.
- an algebraic group is split if it has a split maximal torus
Clearly $\text{GL}_3(\Bbb C)$ has a split maximal torus (for example the group of diagonal matrices). So to compare these definitions, I wanted to check that the standard borel subgroup of upper triangular matrices is split. So I computed it's subnormal series: $$B_3\rhd U_3^2\rhd U_3^1\rhd \{I\}$$ where I denote by $U_n^i$ the upper triangular unipotent matrices, which only have entries on the top $i$ diagonals (so $U_3^1$ is unipotent with just a single entry in the top right position).
Then it seems that $B_3/U_3^2\cong T\cong \Bbb G_m\times\Bbb G_m \times \Bbb G_m$
Easier even $\text{GL}_2(\Bbb C)$ has subnormal series $$B_2\rhd U\rhd \{1\},$$ and it seems $B_2/U\cong T\cong \Bbb G_m\times\Bbb G_m$? So $\text{GL}_2(\Bbb C)$ is not split in the first definition, what am I doing wrong? Surely these are equivalent somehow.
We have (as suggested user Eoin in the comments)
$$B_2=\left\{\begin{pmatrix}a& b \\ 0 & c\end{pmatrix}\right\}\supseteq \left\{\begin{pmatrix}a & b\\ 0 & a\end{pmatrix}\right\}\supseteq \left\{\begin{pmatrix}1 & b\\ 0 & 1\end{pmatrix}\right\}\supseteq \{e\}$$
With quotients $\mathbb{G}_m$, $\mathbb{G}_m$, and $\mathbb{G}_a$ respectively.
The main reason I wanted to comment was, and maybe this doesn't matter to you, but 2. is the 'correct' definition in total generality. The point is that for any smooth connected solvable group $B$ (which Borels are) over a perfect field $k$ one has a decomposition $B\cong R_u(B)\rtimes T$ where $T$ is a torus. Then, it's not hard to see that 1. and 2. are equivalent if you know that $R_u(B)$ admits a normal series with quotients being $\mathbb{G}_a$. A unipotent group $U$ is called split if it admits such a normal series. Over a perfect field every unipotent group is split, but this is false in the imperfect case. So, in summary definitions 1 and 2 are not equivalent over imperfect fields, in which case it is definition 2. which is the correct one.