= Solution
Use the stable finite-dimensional description. Define addition by block sum, $[u]+[v]=[\operatorname{diag}(u,v)]$. Stabilizations and <homotopies> preserve block sums, so this is well-defined. Associativity is immediate after identifying the three blocks; the identity class is zero.
Swapping two blocks is conjugation by a permutation unitary. Every finite-dimensional unitary is joined to the identity, since the matrix spectral theorem writes it as $e^{iA}$ with $A$ self-adjoint. Conjugation along that path gives a <homotopy> swapping the blocks, proving commutativity.
The inverse of $[u]$ is $[u^{-1}]$. To verify it rather than assuming a group structure, first stabilize to equal-sized blocks and put
$$
R_t=\begin{pmatrix}\cos t\,I&-\sin t\,I\\\sin t\,I&\cos t\,I\end{pmatrix}.
$$
The invertible <homotopy>
$$
\operatorname{diag}(u,I)\,R_t\operatorname{diag}(I,u^{-1})R_t^{-1},\qquad0\leq t\leq\pi/2,
$$
starts at $\operatorname{diag}(u,u^{-1})$ and ends at the identity. Thus \b[$\boxed{K^1(X)\text{ is an abelian group}.}$] The constructions are continuous in $x$ and remain valid for maps into $GL_N$; using polar factors also makes them <homotopies> in the unitary model.
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