Decompose complex vector by SU(4)

Question

This question is about to decompose (or reduce dimension) complex vector by $SU\left( 4 \right)$. Given any $4\times1$ complex vector $B$. We can build $a_i$,and matrix$\lambda_i,i=1\ldots n $, $a_i \in\mathbb{R}$,$\lambda_i$ $\in$ one of the basis of generating matrices of $SU\left( 4 \right)$, make $$e^{ja_i\lambda_1}\times e^{ja_2\lambda_2} \times\ldots\times e^{ja_n\lambda_n}\times B= \left[ \begin{matrix} C \\ 0 \\ 0 \\ 0 \\ \end{matrix} \right] $$where $C$ is a complex.

Question is:

1. The maximum value of $n$? (Maybe 6 I think, but I don't know why)
2. How to select such $a_i,\lambda_i$?
3. Where can I find the references to solve this problem.

Many thanks in advance.

Answer 1

You only need one $A_i$, where the first row is just $B^\dagger$ while the other three are row vectors orthogonal to $B$.