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The decomposition terminates because the dimension of the representation space is finite. At each step, a proper nonzero $G$-invariant subspace $W$ is split off, leaving a complementary invariant subspace $W^{\perp}$.

Conditions: $V$ is a finite-dimensional vector space.; $W$ is a proper nonzero $G$-invariant subspace of $V$.; The inner product is $G$-invariant, ensuring $W^{\perp}$ is also $G$-invariant.

The decomposition terminates because the dimension of the representation space is finite. At each step, a proper nonzero $G$-invariant subspace $W$ is split off, leaving a complementary invariant subspace $W^{\perp}$.

Conditions: $V$ is a finite-dimensional vector space.; $W$ is a proper nonzero $G$-invariant subspace of $V$.; The inner product is $G$-invariant, ensuring $W^{\perp}$ is also $G$-invariant.