If I fix a basis $\{e_i\}_{i=1}^n$ and let $A$ be the matrix of a bilinear form with respect to this basis, then it is well known that if I apply a rotation $(e_1',\dots,e_n')=(e_1,\dots,e_n)C$, for some invertible matrix $C$, then the matrix of the bilinear form with respect to $\{e_i'\}_{i=1}^n$ may be written as $C^tAC$.
However, assume that for some $k<n$, $T\in\mathbb{R}^{k\times k}$ is some rotation applied to a subset of the basis $\{e_i\}_{i=1}^n$ corresponding to some index set $R\subset\{1,\dots,n\}$. Thus, to apply the rotation $T$ to the matrix $A$, we would need to appropriately pad identity blocks to produce a larger rotation matrix, call it $\bar{T}$. Where, one construction of $\bar{T}$ might be:
$$\bar{T}:=I_n$$ $$\bar{T}(R):=T$$
My question is to see if there is some common notation for performing such a padding operation when applying a rotation only to a subset of basis elements? If the subset $R=\{i,\dots,j\}$ for some $i<j$, I have seen the notation:
$$\bar{T}:=I_{i-1}\oplus T\oplus I_{n-j}$$
However, this notation can get messy if the indices in $R$ are not consecutive. Any ideas are greatly appreciated, thanks in advance.