Borel subgroup of $Sp(4,\mathbb{C})$

Question

I am trying to understand Borel subgroups of $Sp(4,\mathbb{C})$. I think that the following is a Borel subgroup of $Sp(4, \mathbb{C})$: the subset of $Sp(4, \mathbb{C})$ of all lower triangular matrices $X=\left(\begin{array}{cccc} x_{1,1} & 0 & 0 & 0\\ x_{2,1} & x_{2,2} & 0 & 0\\ x_{3,1} & x_{3,2} & x_{3,3} & 0\\ x_{4,1} & x_{4,2} & x_{4,3} & x_{4,4} \end{array}\right)$ such that $X^T J X = J$, where $J = \left(\begin{array}{cccc} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right)$. Since $X^T J X = J$, we have $$ X^T J X - J = 0 = \\ \left(\begin{array}{cccc} 0 & x_{1,1}\, x_{3,2} + x_{2,1}\, x_{4,2} - x_{2,2}\, x_{4,1} & x_{1,1}\, x_{3,3} + x_{2,1}\, x_{4,3} - 1 & x_{2,1}\, x_{4,4}\\ x_{2,2}\, x_{4,1} - x_{2,1}\, x_{4,2} - x_{1,1}\, x_{3,2} & 0 & x_{2,2}\, x_{4,3} & x_{2,2}\, x_{4,4} - 1\\ 1 - x_{2,1}\, x_{4,3} - x_{1,1}\, x_{3,3} & - x_{2,2}\, x_{4,3} & 0 & 0\\ - x_{2,1}\, x_{4,4} & 1 - x_{2,2}\, x_{4,4} & 0 & 0 \end{array}\right). $$ There are $5$ independent equations above. Therefore $\dim B^- = 10 - 5 = 5$, where $B^-$ is the above Borel subgroup.

But we also have $B^- = U^- T$, where $T$ is the torus. Since $T^T J T = J$, $\dim T=2$. The dimension of $U^-$ is the length of the longest word in $C_2$ Weyl group which is $4$. Therefore $\dim B^- = 4 + 2 = 6 \neq 5$. Is $\dim B^- = 5$ or $6$? Thank you very much.