Decompose real positive-definite symmetric matrix Can any $M\in \mathbb{R}^{n\times n}_{\text{sym}+}$ (real, positive-definite, symmetric matrix) be decomposed in $M=C^{T}.C$ with $C\in \text{GL}_n(R)$ and vice versa ($C\in \text{GL}_n(R)\implies C^{T}.C\in \mathbb{R}^{n\times n}_{\text{sym}+}$)?
This question is related to inner products on real n-dim vector spaces with associated matrices $M\in \mathbb{R}^{n\times n}_{\text{sym}+}$. Under a basis transformation with associated matrix $C\in \text{GL}_n(R)$, the Gram matrix $M$ transforms as follows
$$M_b=C^T\cdot M_e\cdot C$$
If we choose $M_e=\text{id}$ then all other possible Gram matrices $M_b$ can be constructed by changing the basis so that $M_b=C^T\cdot C$ (at least if the statement in the beginning is true). If this is true, is their a way to characterize a real inner product on the basis of its Gram matrix? Maybe the basis choice for which $M_e=\text{id}$? And if so, when is this choice Euclidean?

I guess you're thinking of $M$ as the Gram matrix for a particular basis (which is just the matrix for the inner product as a symmetric bilinear form). There is an orthonormal basis on $\mathbb R^n$ with respect to this inner product, and so, yes, you can change basis to make $M$ turn into the identity matrix. (BTW, this works with any nondegenerate symmetric form, not necessarily positive-definite.)
But, to answer your more general question, for any $C\in GL_n(\mathbb R)$, the matrix $C^\top C$ is symmetric (just take its transpose) and positive-definite ($C^\top Cx\cdot x = Cx\cdot Cx  = \|Cx\|^2\ne 0$, since $C$ is nonsingular). Conversely, by the Spectral Theorem, there is an orthonormal basis for $\mathbb R^n$ consisting of eigenvectors of your given $M$. Moreover, since $M$ is positive definite, every eigenvalue is positive (see the calculation just above), and so $M=Q \Lambda Q^\top$ for some orthogonal matrix $Q$ and some diagonal matrix $\Lambda$ with positive entries. Now take $C=Q\sqrt\Lambda Q^\top$, where $\sqrt\Lambda$ denotes the diagonal matrix whose entries are the (positive) square roots of the entries of $\Lambda$. $C$ is a symmetric matrix with $C^2 = M$.
