- #1

- 713

- 5

## Main Question or Discussion Point

Hi,

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 a

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).

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 a

*subalgebra*of ##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).