Let $X \sim N(0,1)$ and B such that $P(B=1)=P(B=-1)=1/2$, two independent variables. Define - Y=BX.
Then Y is also Gaussian since -
$$F_{Y}(y)=P(Y\leq y)=P(BX \leq y)=P(B=1)P(BX \leq y|B=1)+P(B=-1)P(BX \leq y|B=-1)=P(B=1)P(X \leq y|B=1)+P(B=-1)P(-X \leq y|B=-1)=P(B=1)P(X \leq y)+P(B=-1)P(-X \leq y)=\frac{1}{2}[F_{X}(y)+F_{-X}(y)]$$
Define Z = -X, Since $X$ Gaussian then also $Z$, and $$E[Z]=E[-X]=-E[X]=0$$ and $$Var(Z)=Var(-X)=(-1)^2 Var(X)= Var(X)$$
Hence also $Z \sim N(0,1)$ and particulaty $F_{Z}(y)=F_{X}(y)$
Yields - $$F_{Y}(y)=\frac{1}{2}[F_{X}(y)+F_{Z}(y)]=F_{X}(y)$$
Now X and Y are uncorrelated - $$E[X]=E[Y]=0$$ $$E[XY]=E[BX^2]=E[B]E[X^2]=0 \cdot E[X^2]=0$$
We know that if two Gaussian are uncorrelated, they are also independent, and if two Gaussian are independent, they are "jointly Gaussian".
Hence, every linear combination of X and Y should be also Gaussian.
But, W =Y+X isn't Gaussian since $P(W=0)=1/2$ $$P(W=0)=P(B=1)P(W = 0 |B=1)+P(B=-1)P(W = 0|B=-1)=\frac{1}{2}[P(X+Y=0|B=1)+P(X+Y=0|B=-1)]=\frac{1}{2}[P(2X=0|B=1)+P(X-X=0|B=-1)]=\frac{1}{2}$$
So W takes a specific value with non-zero probability which means it isn't even a continuous and particularly not Gaussian.
Where is my mistake?