Finding orthonormal basis within the Aitchison simplex geometry Inside $\mathbb{R}^n$, consider the space
$$S_\kappa = \{ x \in \mathbb{R}^n \ | \ \sum_{i=1}^n x_i = \kappa, \ \text{and} \ \ x_i \in (0,1)  \ \text{for all} \ i \in \{1,..,n\}\}.$$
On this simplex, one can define the Aitchison geometry with pertubation (addition), powering (multiplication), and a scalar product (-> norms, distances), which turns it into a "vector space". You probably need to know this geometry in order to answer this question.
Anyways, within that geometry, how do I go about finding an orthonormal basis, i.e, if $\oplus$ and $\odot$ denote pertubation and powering, then we want to find $e_i$ such that for any $x$ in $S_\kappa$, $x = \bigoplus_{i=1}^n \left( \lambda_i \odot e_i \right)$, where $e_i$ are the basis vectors in $S_\kappa$ which have unit norm and are orthogonal (using the Aitchison scalar product), and $\lambda$s are real constants.
 A: there is (as far as I know) not one canonical way to construct an orthonormal basis for the simplex
\begin{align*}
 S_\kappa = \{\mathbf{x} \in \mathbb{R}^d : \sum^{d}_{i=1} x_i = \kappa, x_i > 0\}.
\end{align*}
I'll therefore outline the perhaps most (more) common construction via an isometry (keyword ilr).
To make the answer sort of self contained I'll denote by $\Vert \mathbf{x} \Vert_1 = \sum_{i=1}^d \vert x_i \vert$ the usual $1$-norm on $\mathbb{R}^d$ (which will here simply be the sum as we are dealing with vectors with positive entries), and $C$ denotes the closure of a vector $\mathbf{x} \in (0,\infty)^d$ defined as
\begin{align*}
C[\mathbf{x}] = \left(\frac{\kappa}{\Vert \mathbf{x} \Vert_1}x_1,\ldots,\frac{\kappa}{\Vert \mathbf{x} \Vert_1}x_d\right).
\end{align*}
For $\mathbf{x},\mathbf{y} \in S_{\kappa}$ and $\alpha \in \mathbb{R}$ I'll use the operations
\begin{align*}
 \mathbf{x}\oplus\mathbf{y} &:= C[(x_1 y_1,\ldots,x_d y_d)],\\
 \alpha \odot \mathbf{x} &:= C[(x_1^\alpha,\ldots,x_d^\alpha)].
\end{align*}
As you already noted, $(S_{\kappa},\oplus,\odot)$ is a $\mathbb{R}$ vector space, and of course for $\mathbf{x} \in S_{\kappa}$ we have $C[\mathbf{x}] = \mathbf{x}$.
The standard inner product on $\mathbb{R}^d$ is denoted by $\langle , \rangle_{\mathbb{R}^d}$, while the inner product on $S_\kappa$ is given by
\begin{align*}
 \langle \mathbf{x},\mathbf{y} \rangle_A = \frac{1}{2D} \sum^{d}_{i=1} \sum^{d}_{j=1} \ln \frac{x_i}{x_j} \ln \frac{y_i}{y_j}. 
\end{align*}
Coming back to the construction of our basis we also need a vector sub-space of $\mathbb{R}^d$ given as
\begin{align*}
 V = \{\mathbf{v} \in \mathbb{R}^d : \sum^{d}_{i=1} v_i = 0\}.
\end{align*}
You'll notice that with respect to the standard inner product on $\mathbb{R}^d$ these are exactly the vectors that are orthogonal to $(1,\ldots,1)$.
For $\mathbf{x} \in S_\kappa$ we can further consider the geometric mean
\begin{align*}
g(\mathbf{x}) = \left( \prod_{i=1}^d x_i \right)^{1/d}.
\end{align*}
This allows us to define the centered log-ratio (clr) transform for $\mathbf{x} \in S_\kappa$ as
\begin{align*}
 \mbox{clr} \colon S_\kappa \to V, \quad \mathbf{x} \mapsto \mbox{clr}(\mathbf{x}) = \left( \ln \frac{x_1}{g(\mathbf{x})},\ldots,\ln \frac{x_{d}}{g(\mathbf{x})} \right).
\end{align*}
clr also has an inverse
\begin{align*}
\mbox{clr}^{-1} \colon V \to S_\kappa, \quad \mathbf{v} \mapsto C[\exp(v_1),\ldots,\exp(v_d)],
\end{align*}
and a number of good properties: it is linear, i.e., $\mbox{clr}(\alpha \odot \mathbf{x} \oplus \mathbf{y}) = \alpha \mbox{clr}(\mathbf{x}) + \mbox{clr}(\mathbf{y})$, and it is an isometry:
\begin{align*}
 \left\langle \mathbf{x},\mathbf{y} \right\rangle_A = \left\langle \mbox{clr}(\mathbf{x}),\mbox{clr}(\mathbf{y}) \right\rangle_{\mathbb{R}^d}.
\end{align*}
The fact that it is an isometry can now be used to construct an orthonormal basis for $S_\kappa$.
For a given orthonormal basis $\{\mathbf{v}_1,\ldots,\mathbf{v}_{d-1}\}$ of $V$ (which is just a linear sub-space of $\mathbb{R}^d$) we can now define an orthonormal basis for $S_\kappa$ via
\begin{align*}
 \{\mbox{clr}^{-1}(\mathbf{v}_1),\ldots,\mbox{clr}^{-1}(\mathbf{v}_{d-1})\}.
\end{align*}
Here you see that you will get (potentially) different bases for $S_\kappa$ depending on $\{\mathbf{v}_1,\ldots,\mathbf{v}_{d-1}\}$.
Following this approach now leads to the so called isometric log-ratio (ilr) coordinates for a vector $\mathbf{x} \in S_\kappa$ given as
\begin{align*}
 \mbox{ilr}(\mathbf{x}) = (\langle \mathbf{x},\mbox{clr}^{-1}(\mathbf{v}_1)\rangle_A,\ldots,\langle\mathbf{x},\mbox{clr}^{-1}(\mathbf{v}_{d-1})\rangle_A).
\end{align*}
What you see is that the coordinates depend on the chosen basis of $V$.
Furthermore ilr is an isometric isomorphism, i.e., for $\mathbf{x},\mathbf{y} \in S_\kappa$ and $\alpha \in \mathbb{R}$ we get
\begin{align*}
 \mbox{ilr}(\alpha \odot \mathbf{x} \oplus \mathbf{y}) &= \alpha\mbox{ilr}(\mathbf{x}) + \mbox{ilr}(\mathbf{y}),\\
 \langle \mathbf{x},\mathbf{y} \rangle_A &= \langle \mbox{ilr}(\mathbf{x}),\mbox{ilr}(\mathbf{y}) \rangle_{\mathbb{R}^d}.
\end{align*}
With the ilr-coordinates you therefore indeed have the representation
\begin{align*}
 \mathbf{x} = \oplus_{i=1}^{d-1} \mbox{ilr}(\mathbf{x})_i \odot \mbox{clr}^{-1}(\mathbf{v}_{i}).
\end{align*}
In this setting the question at hand is therefore maybe not how to construct an orthonormal basis for $S_\kappa$, but how to construct such a basis that is interpretable - after all we are interested in this construction because we want to do statistics and interpret the results.
For an answer/discussion to this questions I would however refer you to the literature:
The paper (although about a more specific topic)
Kynčlová, Filzmoser, Hron (2015). Modeling Compositional Time Series with Vector Autoregressive Models. Journal of Forecasting, J. Forecast. 34, 303–314
contains a nice overview on the basis construction while you can find more details about (choices of) orthonormal bases (also how to construct them with Gram-Schmidt for $V$) in
J. J. Egozcue, V. Pawlowsky-Glahn, G. Mateu-Figueras, C. Barceló-Vidal (2003). Isometric Logratio Transformations for Compositional Data Analysis. Mathematical Geology, Vol. 35, No. 3.
Finally a nice general introduction to the subject (including a chapter on orthogonal bases) is given in the lecture notes
http://www.compositionaldata.com/material/others/Lecture_notes_11.pdf
