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consider a (finite dimensional) vector space ##V=U\oplus W##, where the subspaces ##U## and ##V## are not necessarily orthogonal, equipped with a bilinear product ##*:V\times V \rightarrow V##.

The subspace ##U## is closed under multiplication ##*##, thus ##U## is asubalgebraof ##V##.

Does this imply that also ##W## is a subalgebra of ##V##?

(Note, I can already prove the special case that if U and W are orthogonal, then both U and W are indeed subalgebras).

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# A Subalgebras and direct sums

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