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# Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket = XY − YX, because the Lie bracket of any two derivations is a derivation.
# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
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# the compact form, sp ( n ), which is the Lie algebra of Sp ( n ),
# the normal form ( or split form ), sp ( 2n, R ), which is the Lie algebra of Sp ( 2n, R ).
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:: Chapter: Exterior algebra and differential calculus # 6 in 1st ed, # 7 in 2nd.
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