This is probabably a very silly question, stemming from some fundamental misunderstanding I have of the relevant definitions, but I am stumped by it.
I know that any two Cartan subalgebras of $\mathfrak{sl}(2,\mathbb{C})$ should be conjugate. I understand this to mean in particular that there is $g\in\mathrm{SL}(2,\mathbb{C})$ such that $$\mathrm{Ad}(g)\cdot \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}= g \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} g^{-1}= \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}. $$
This seems impossible, though, as these matrices have different characteristic polynomials.
What have I misunderstood?
It is due to the fact that $Ad(g)\pmatrix{1& 0\cr 0&-1}$ maybe $i\pmatrix{0 &1\cr -1&0}$ which has the same characteristic polynomial than $\pmatrix{1& 0\cr 0&-1}$
take $g={\sqrt2\over 2}\pmatrix{1&-i\cr -i&1}$.