= Conjugacy of Cartan subalgebras of the symplectic Lie algebra
{title2=$g\mathfrak h g^{-1}=\mathfrak h',\quad g\in\operatorname{Sp}(V)$}
Over an <algebraically closed field> of <characteristic zero>, any two <Cartan subalgebras> of the <symplectic Lie algebra> are conjugate by the <symplectic group>. A <Cartan subalgebra> $\mathfrak h$ is an <abelian Lie algebra> and equals its <centralizer of a Lie subalgebra>. The <symplectic centralizer decomposition> therefore forces all its <weight spaces> in $V$ to be one-dimensional, nonzero, and paired by opposite <weights of a representation>. Choose a <basis vector> in each space, normalizing the <perfect pairing> with its opposite to one. These vectors form a <symplectic basis> $\mathcal B$. In this <basis>, $\mathfrak h$ is contained in the <Lie subalgebra> of <diagonal matrices> with entries $(t_1,\ldots,t_n,-t_1,\ldots,-t_n)$. All those <diagonal matrices> commute with $\mathfrak h$, so equality with its <centralizer of a Lie subalgebra> forces equality with the entire diagonal <Lie subalgebra>. Repeating for $\mathfrak h'$ gives a second <symplectic basis>. The <linear map> sending one <symplectic basis> to the other belongs to the <symplectic group> and conjugates the two diagonal <Lie subalgebras>, proving the claim.
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