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Direct matrix multiplication gives the multiplication tableTogether with $I$ acting as the identity, this shows that the product of any two real linear combinations is again a real linear combination. Thus $\Omega$ is closed under multiplication. It is the complex matrix representation of quaternions.

Its general element has the formIf this matrix is zero, real and imaginary parts of its first row give $a=b=c=d=0$. Hence $I,J,K,L$ are linearly independent over $\mathbb{R}$ and form a basis:Put $v=bJ+cK+dL$. Since distinct imaginary units anticommute, the mixed terms in $v^2$ cancel, leaving $v^2=-(b^2+c^2+d^2)I$. Scalars commute with $v$, soThe reversed product is the same. The coefficient is strictly positive for every nonzero element. ConsequentlyThis also identifies $\Omega$ as a real noncommutative division algebra, not as the full algebra of complex two-by-two matrices.