Unitary and special unitary Lie groups

Regard $M_n(\mathbb{C})$ as ${\mathbb{R}}^{2n^2}$. The complex determinant is a finite polynomial in matrix entries by [F5], hence its real and imaginary parts are real polynomials. By [F2], ${\mathrm{GL}}_n(\mathbb{C})=\{A:\det A\mathrm{\ne }0\}$ is open in this real vector space; multiplication is polynomial and inversion is real smooth there by the adjugate formula. Thus it is a real Lie group by [F1]. Its left-invariant field with identity value $X$ is $A\mapsto AX$; differentiating these linear fields gives the tangent bracket $[X,Y]=XY-YX$ in the convention of [F1]. Now $F(A)=A^{\ast }A$ maps this open group smoothly into the real vector space of Hermitian matrices and has $d{F}_A(X)=X^{\ast }A+A^{\ast }X$. For Hermitian $S$, $X=\frac{1}{2}{A}^{-\ast }S$ maps to $S$, so [F4] makes $U(n)=F^{-1}(I)$ embedded; the adjoint-product identities make it a subgroup. At $I$ its tangent kernel is $X^{\ast }+X=0$.

For $A\in U(n)$, $\mathrm{\mid }\det A{\mathrm{\mid }}^2=\det (A^{\ast }A)=1$, so determinant maps $U(n)$ into the unit circle. Near $1$ that circle has the real coordinate $z\mapsto \mathrm{Im}z$ on the arc $\mathrm{Re}z>0$. Differentiating the finite determinant formula at $I$ gives $d(\det {)}_I(X)=\mathrm{tr}X$; on skew-Hermitian $X$ this is imaginary and every imaginary scalar occurs from a diagonal $X$. Left multiplication by any $A\in SU(n)$ transports this surjectivity to $A$. Hence $1$ is a regular value of the circle-valued determinant map and [F4] makes $SU(n)$ embedded in $U(n)$, with tangent kernel $\mathrm{tr}X=0$ at $I$. Its subgroup operations are smooth by restriction.

Both tangent spaces are closed under commutator: adjoint reverses products, and trace of a commutator vanishes by finite reindexing. Hence they are the asserted Lie algebras.

At $n=1$, $\mathfrak{u}(1)=i\mathbb{R}$ and $\mathfrak{s}\mathfrak{u}(1)=0$; $n=0$ is excluded by the Statement. No interval, endpoint, arbitrary metric choice, or biconditional occurs. The ambient determinant is nonzero exactly on ${\mathrm{GL}}_n(\mathbb{C})$ by [F2]. ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited through [F1]; finite coordinates add no choice.