feat(site/ai-elements): add background process indicator to execute tool

Show a LayersIcon with a "Running in background" tooltip on execute
tool calls that returned a background_process_id. The icon sits in
the header controls area next to the spinner/copy button and is
always visible (not hover-gated) since it conveys meaningful status.

Detection is purely derived from the existing result payload — no
polling or new state management required.

site/src/components/ai-elements/tool.stories.tsx

@@ -51,6 +51,21 @@ export const ExecuteSuccess: Story = {
 	},
 };
 
+export const ExecuteBackgrounded: Story = {
+	args: {
+		result: {
+			output: "",
+			background_process_id: "proc-abc-123",
+		},
+	},
+	play: async ({ canvasElement }) => {
+		const canvas = within(canvasElement);
+		expect(canvas.getByText("$ git fetch origin")).toBeTruthy();
+		// The layers icon should be present as the background indicator.
+		expect(canvasElement.querySelector(".lucide-layers")).not.toBeNull();
+	},
+};
+
 export const ExecuteAuthRequired: Story = {
 	args: {
 		result: {

site/src/components/ai-elements/tool/ExecuteTool.tsx

@@ -3,6 +3,7 @@ import {
 	ChevronDownIcon,
 	CircleAlertIcon,
 	ExternalLinkIcon,
+	LayersIcon,
 	LoaderIcon,
 	TriangleAlertIcon,
 } from "lucide-react";
@@ -12,6 +13,11 @@ import { cn } from "utils/cn";
 import { Button } from "#/components/Button/Button";
 import { CopyButton } from "#/components/CopyButton/CopyButton";
 import { ScrollArea } from "#/components/ScrollArea/ScrollArea";
+import {
+	Tooltip,
+	TooltipContent,
+	TooltipTrigger,
+} from "#/components/Tooltip/Tooltip";
 import {
 	BORDER_BG_STYLE,
 	COLLAPSED_OUTPUT_HEIGHT,
@@ -28,7 +34,8 @@ export const ExecuteTool: React.FC<{
 	output: string;
 	status: ToolStatus;
 	isError: boolean;
-}> = ({ command, output, status }) => {
+	isBackgrounded?: boolean;
+}> = ({ command, output, status, isBackgrounded = false }) => {
 	const [expanded, setExpanded] = useState(false);
 	const outputRef = useRef<HTMLPreElement | null>(null);
 	const hasOutput = output.length > 0;
@@ -60,6 +67,14 @@ export const ExecuteTool: React.FC<{
 					{isRunning && (
 						<LoaderIcon className="h-3.5 w-3.5 shrink-0 animate-spin motion-reduce:animate-none text-content-secondary" />
 					)}
+					{isBackgrounded && !isRunning && (
+						<Tooltip>
+							<TooltipTrigger asChild>
+								<LayersIcon className="h-3.5 w-3.5 shrink-0 text-content-secondary" />
+							</TooltipTrigger>
+							<TooltipContent>Running in background</TooltipContent>
+						</Tooltip>
+					)}
 					<span className="opacity-0 transition-opacity group-hover/exec:opacity-100">
 						<CopyButton text={command} label="Copy command" />
 					</span>

site/src/components/ai-elements/tool/Tool.tsx

@@ -102,6 +102,8 @@ const ExecuteRenderer: FC<ToolRendererProps> = ({
 		rec ? asString(rec.provider_type).trim() : "",
 	);
 
+	const backgroundProcessId = rec ? asString(rec.background_process_id) : "";
+
 	if (authRequired && authenticateURL) {
 		return (
 			<ExecuteAuthRequiredTool
@@ -118,6 +120,7 @@ const ExecuteRenderer: FC<ToolRendererProps> = ({
 			output={output}
 			status={status}
 			isError={isError}
+			isBackgrounded={backgroundProcessId.length > 0}
 		/>
 	);
 };