Let \documentclass[12pt]{minimal}
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$$\mathfrak{g}$$
\end{document} be a reductive Lie algebra over an algebraically closed field of characteristic zero and \documentclass[12pt]{minimal}
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$$\mathfrak{g} = \mathfrak{g}_0\oplus \mathfrak{g}_1 $$
\end{document} an arbitrary \documentclass[12pt]{minimal}
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$$\mathbb{Z}_2 $$
\end{document}-grading. We consider the variety \documentclass[12pt]{minimal}
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$$\mathfrak{C}_1= \{ (x,y)|[x,y] = 0\}\subset \mathfrak{g}_1\times \mathfrak{g}_1 $$
\end{document}, which is called the commuting variety associated with the \documentclass[12pt]{minimal}
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$$\mathbb{Z}_2 $$
\end{document}-grading. Earlier it was proved by the author that \documentclass[12pt]{minimal}
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$$\mathfrak{C}_1 $$
\end{document} is irreducible, if the \documentclass[12pt]{minimal}
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$$\mathbb{Z}_2 $$
\end{document}-grading is of maximal rank. Now we show that \documentclass[12pt]{minimal}
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$$\mathfrak{C}_1 $$
\end{document} is irreducible for \documentclass[12pt]{minimal}
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$$(\mathfrak{g},\mathfrak{g}_0 ) = (\mathfrak{s}\mathfrak{l}_{2n} ,\mathfrak{s}\mathfrak{p}_{2n} )$$
\end{document} and (E6,F4). In the case of symmetric pairs of rank one, we show that the number of irreducible components of \documentclass[12pt]{minimal}
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$$\mathfrak{C}_1 $$
\end{document} is equal to that of nonzero non-ϑ-regular nilpotent G0-orbits in \documentclass[12pt]{minimal}
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$$\mathfrak{g}_1 $$
\end{document}. We also discuss a general problem of the irreducibility of commuting varieties.
