= Solution
A <Lie group> is a <group> and <smooth manifold> whose multiplication and inversion are smooth. A <Lie algebra> is a <vector space> with a bilinear alternating bracket satisfying the <Jacobi identity>. For a <Matrix Lie group> $G\subset GL(m,\mathbb C)$, a <logarithmic chart of a matrix Lie group> near $g$ sends $h$ to $\log(g^{-1}h)\in\mathfrak g=T_I G$; the <matrix exponential> is its local inverse, and the <Baker--Campbell--Hausdorff formula> makes the local group operations smooth.
For $B_1,B_2\in\mathfrak g$, consider the group commutator
$$
C(s,t)=e^{sB_1}e^{tB_2}e^{-sB_1}e^{-tB_2}\in G.
$$
Its logarithm takes values in the vector space $\mathfrak g$, and expansion at $(0,0)$ gives
$$
\frac{\partial^2}{\partial s\,\partial t}\bigg|_{(0,0)}\log C(s,t)
=B_1B_2-B_2B_1.
$$
Thus $\mathfrak g$ is closed under the <commutator>. Bilinearity, alternation, and the Jacobi identity follow from matrix multiplication, so this proves that the <Lie algebra of a matrix Lie group> has
$$
\boxed{[B_1,B_2]=B_1B_2-B_2B_1.}
$$
A <principal bundle> $\pi:P\to M$ with structure group $G$ is a smooth <fiber bundle> with a free right $G$-action, each fiber a single orbit, and equivariant local trivializations $\pi^{-1}(U)\cong U\times G$.
For the right action of the <unitary group> on $GL(n,\mathbb C)$, define
$$
q(g)=\frac12\log(gg^*)\in H(n).
$$
The matrix $gg^*$ is a <positive-definite matrix> and a <Hermitian matrix>, and $q(gu)=q(g)$ for $u\in U(n)$. The <polar decomposition of an invertible complex matrix> gives the unique factorization
$$
g=e^{q(g)}u(g),
\qquad u(g)=e^{-q(g)}g\in U(n).
$$
Consequently
$$
\Phi:H(n)\times U(n)\longrightarrow GL(n,\mathbb C),
\qquad(B,u)\longmapsto e^Bu
$$
is a diffeomorphism. If $W\subset H(n)$ is a neighborhood of zero and $N=q^{-1}(W)$, then $N$ is an open neighborhood of $I$, it is a union of complete orbits, and
$$
V=\bigcup_{h\in N}R(h)=N\cong W\times U(n).
$$
Moreover, two matrices have the same value of $q$ exactly when they differ by right multiplication by a unitary matrix. Thus $q$ induces the smooth identification
$$
\boxed{M=GL(n,\mathbb C)/U(n)\cong H(n),}
$$
under which $\pi$ is projection $H(n)\times U(n)\to H(n)$. This is the <general linear group modulo the unitary group>, and $\pi$ is in fact a globally trivial principal $U(n)$-bundle.
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