The Lie algebra of a closed subgroup is a linear subspace

Idea. The sum $X+Y$ is approximated through products of elements of $H$, and $H$ is closed.

Contains $0$. $e^{t\cdot0}=I\in H$, so $0\in\mathfrak h$.

Scalar multiples. If $X\in\mathfrak h$ and $c\in\mathbb{R}$, then for every $t$: $e^{t(cX)}=e^{(tc)X}\in H$ because $tc$ is just another real number. So $cX\in\mathfrak h$.

Sums. Let $X,Y\in\mathfrak h$ and fix $t\in\mathbb{R}$. For every integer $k\ge1$, $e^{tX/k}\in H$ and $e^{tY/k}\in H$ ($H$ contains $e^{sX}$ for every real $s$), hence the product
$$g_k=\bigl(e^{tX/k}e^{tY/k}\bigr)^k\in H.$$
By the Lie product formula applied to $A=tX$, $B=tY$, $g_k\to e^{t(X+Y)}$ as $k\to\infty$. The limit is invertible (it is an exponential), so it lies in $GL_n$, and $H$ is closed in $GL_n$: $e^{t(X+Y)}\in H$. This holds for every $t$, so $X+Y\in\mathfrak h$.

Remark. The hypothesis "closed" is used in the last step. For the non-closed subgroup $\varphi(\mathbb{R})$ of the torus in $SO(4)$ the limit $e^{t(X+Y)}$ need not stay in the subgroup: its closure is the whole torus, whose Lie algebra is two-dimensional, larger than the line generated by the winding direction.