= Simple bivector
{title2=$B=u\wedge v$}
= Decomposable bivector
{synonym}
A <simple bivector> can be expressed as one <exterior product> of two <vectors>. Changing the pair need not change its area element: $u\wedge(v+cu)=u\wedge v$ for every <scalar> $c$, by <bilinearity> and $u\wedge u=0$. This operation shears the represented <parallelogram> while preserving its oriented area.
Every <bivector> in a real vector space of <dimension> at most three is simple, but this fails in <dimension> four. For an <ordered basis> $e_1,\ldots,e_4$, consider
$$
B=e_1\wedge e_2+e_3\wedge e_4.
$$
The two components describe independent coordinate planes. Their square is
$$
B\wedge B=2e_1\wedge e_2\wedge e_3\wedge e_4\ne0,
$$
whereas $(u\wedge v)\wedge(u\wedge v)=0$ because the factors repeat. Thus this $B$ cannot be one <simple bivector>. Over the <real numbers> in <dimension> four, the converse also holds: a nonzero <bivector> with $B\wedge B=0$ is simple. The nonzero factor two is important; that criterion cannot be applied unchanged in <characteristic two>.
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