= Solution
The <spin> <Hilbert space> is a <tensor product> of two two-dimensional spaces and therefore has <dimension> four. <Angular momentum addition> decomposes it as
$$
\boxed{S=1:\ M_S=1,0,-1;\qquad S=0:\ M_S=0.}
$$
These are three <orthogonal> <triplet states> and one <singlet state>. For identical <electrons> the \b[complete] <wavefunction> must also satisfy <fermionic exchange symmetry>: a symmetric triplet <spin> factor accompanies an antisymmetric spatial factor, whereas the antisymmetric singlet <spin> factor accompanies a symmetric spatial factor. If both <electrons> occupy the same <spatial orbital>, only the singlet <spin> factor is allowed. The four-dimensional <spin> space does not assert four allowed <spin> states for that particular spatial occupation.
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