= Symplectic centralizer decomposition
{title2=$C_{\mathfrak{sp}(V)}(\mathfrak a)\cong\mathfrak{sp}(V_0)\oplus\bigoplus_{\{\lambda,-\lambda\}}\mathfrak{gl}(V_\lambda)$}
Let $\mathfrak a$ be a <toral subalgebra> and an <abelian Lie algebra> in the <symplectic Lie algebra>, over an <algebraically closed field> of <characteristic zero>. Its <defining representation> has a <weight decomposition>. The <symplectic Lie algebra> identity gives $(\lambda(a)+\mu(a))\omega(v,w)=0$ for $v\in V_\lambda$, $w\in V_\mu$, so $\omega(V_\lambda,V_\mu)=0$ unless $\mu=-\lambda$. Since $\omega$ is a <nondegenerate bilinear form>, its restrictions give a <perfect pairing> between $V_\lambda$ and $V_{-\lambda}$, and a <nondegenerate bilinear form> on $V_0$.
An element of the <centralizer of a Lie subalgebra> preserves each <weight space>. On $V_0$ it is any element of $\mathfrak{sp}(V_0)$. On each nonzero pair, its action $A$ on $V_\lambda$ is arbitrary, and the <perfect pairing> uniquely forces the action on $V_{-\lambda}$ to be the negative <dual map> $-A^*$. These independent choices preserve <Lie brackets>, proving the displayed <direct sum of Lie algebras>; the sum runs over one representative per nonzero pair.
This <centralizer of a Lie subalgebra> is an <abelian Lie algebra> precisely when $V_0=0$ and all the nonzero <weight spaces> are one-dimensional. Indeed, $\mathfrak{gl}_m$ is nonabelian for $m\geq2$ by $[E_{11},E_{12}]=E_{12}$, while every nonzero <symplectic vector space> contains a two-dimensional <symplectic subspace> on which $[\operatorname{diag}(1,-1),E_{12}]=2E_{12}$. In the stated case the surviving summands are copies of $\mathfrak{gl}_1$ and commute. Repeated <weights of a representation>, including a zero one, are therefore exactly what obstructs this commutativity.
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